Bond portfolio optimization

Lecture Notes in Economicsand Mathematical Systems 605

Founding Editors:

M. BeckmannH.P. Künzi

Managing Editors:

Prof. Dr. G. FandelFachbereich WirtschaftswissenschaftenFernuniversität HagenFeithstr. 140/AVZ II, 58084 Hagen, Germany

Prof. Dr. W. TrockelInstitut für Mathematische Wirtschaftsforschung (IMW)Universität BielefeldUniversitätsstr. 25, 33615 Bielefeld, Germany

Editorial Board:

A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten

Michael Puhle

Bond PortfolioOptimization

123

Dr. Michael PuhleAllianz Global Investors Kapitalanlagegesellschaft mbHNymphenburger Straße 112-11680636 [email protected]

Doctoral thesis, University of Passau, 2007

ISBN 978-3-540-76592-9 e-ISBN 978-3-540-76593-6

DOI 10.1007/978-3-540-76593-6

Lecture Notes in Economics and Mathematical Systems ISSN 0075-8442

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Acknowledgements

I would like to express my gratitude to a number of people. First of all, I thankmy thesis advisor Prof. Dr. Jochen Wilhelm for supervising and guiding meduring my years at the chair of finance at Passau University. Prof. Dr. JanosSzaz gave me the opportunity to present previous versions at seminars inBudapest. He also provided valuable feedback and agreed to be the secondreferee.

Prof. Dr. Bernhard Nietert took the time to discuss several parts of thisthesis with me. Dipl.-Kffr. Marion Trautbeck-Kim and Dipl.-Kfm. AndreasKremer read a draft of this thesis on short notice. Furthermore, I would liketo thank Dr. Wolfram Peters for giving me the opportunity to complete thisthesis during my first months at Allianz Global Investors.

My parents and my brother supported this project from the beginningand provided warm encouragement during difficult days. Last but not least, Iwould like to thank my fiancee Veronika for her support especially during thefinal stages.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI

Commonly Used Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Bond Market Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Characteristics of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Term Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Estimating the Term Structure of Interest Rates . . . . . . . . . . . . . 92.5 Classical Theories of the Term Structure of Interest Rates . . . . 102.6 Arbitrage-Free Term Structure Theories . . . . . . . . . . . . . . . . . . . . 112.7 Empirical Properties of the Term Structure of Interest Rates . . 11

3 Term Structure Modeling in Continuous Time . . . . . . . . . . . . . 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Interest Rate Modeling Approaches . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Heath/Jarrow/Morton (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Dynamics of Traded Securities . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Arbitrage-Free Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.4 Excursus: The HJM Drift Condition . . . . . . . . . . . . . . . . . 203.3.5 The Short Rate of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.6 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Vasicek (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.2 Derivation of Zero-Coupon Bond Prices . . . . . . . . . . . . . . 233.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

VIII Contents

3.5 Hull/White (1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.2 Derivation of Zero-Coupon Bond Prices . . . . . . . . . . . . . . 313.5.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Static Bond Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Static Bond Portfolio Selection in Theory . . . . . . . . . . . . . . . . . . . 41

4.2.1 A Short Review of Modern Portfolio Theory . . . . . . . . . . 414.2.2 Application to Bond Portfolios . . . . . . . . . . . . . . . . . . . . . . 434.2.3 Obtaining the Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.4 One-Factor Vasicek (1977) Model . . . . . . . . . . . . . . . . . . . . 514.2.5 Two-Factor Hull/White (1994) Model . . . . . . . . . . . . . . . . 60

4.3 Static Bond Portfolio Selection in Practice . . . . . . . . . . . . . . . . . . 664.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Active Bond Portfolio Selection Strategies . . . . . . . . . . . . 674.3.3 Passive Bond Portfolio Selection Strategies . . . . . . . . . . . 774.3.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Dynamic Bond Portfolio Optimization in Continuous Time 855.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Bond Portfolio Selection Problem in a HJM Framework . . . . . . 87

5.2.1 Dynamics of Prices and Wealth . . . . . . . . . . . . . . . . . . . . . 875.2.2 The Hamilton/Jacobi/Bellman Equation . . . . . . . . . . . . . 895.2.3 Derivation of Optimum Portfolio Weights . . . . . . . . . . . . . 915.2.4 The Value Function for CRRA Utility Functions . . . . . . 94

5.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3.1 One-Factor Vasicek (1977) Model . . . . . . . . . . . . . . . . . . . . 965.3.2 Two-Factor Hull/White (1994) Model . . . . . . . . . . . . . . . . 100

5.4 International Bond Investing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.4.3 Derivation of the Optimum Portfolio Weights . . . . . . . . . 1095.4.4 Interpretation of the Optimum Portfolio Weights . . . . . . 1115.4.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A Heath/Jarrow/Morton (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A.1 Dynamics of Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 119A.2 Arbitrage-Free Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.3 HJM Drift Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.4 Special Case: Hull/White (1994) . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Contents IX

B Dynamic Bond Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . 123

C Dynamic Bond Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . 125C.1 Vasicek (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.2 Hull/White (1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.3 International Bond Portfolio Selection . . . . . . . . . . . . . . . . . . . . . . 126

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Abbreviations

CIR Cox/Ingersoll/Ross (1985)CRRA Constant relative risk aversionEUR Currency code for the EuroHJB Hamilton-Jacobi-Bellman equationHJM Heath/Jarrow/Morton (1992)HW2 Hull/White (1994)InvG Investmentgesetz (German Investment Act)MA Martingale approachMMA Money market accountODE Ordinary differential equationPDE Partial differential equationRRA Relative risk aversionSCA Stochastic control approachSDE Stochastic differential equationUSD Currency code US Dollare.g. Exempli gratia, for exampleff. And following pagesi.e. Id est, that ismax Maximizemin Minimizep. Pagepp. Pagess.t. Subject tostd. Standard deviationw.r.t. With respect to

Commonly Used Symbols

t TimeT Maturity date or investment horizonR(t, T ) Continuously-compounded spot interest rate from t to TP (t, T ) Price at time t for a zero-coupon bond with maturity date Tµ Drift of the zero-coupon bond priceσi i-th volatility of the zero-coupon bond price, 1 ≤ i ≤ dλi i-th market price of interest rate risk, 1 ≤ i ≤ dr(t) Short rate of interest, r(t) = f(t, t)α Drift of the short rateσr Volatility of the short ratetR(T, τ) Forward interest rate set at time t for a loan that starts at time

T and is to be repaid at time τf(t, T ) Instantaneous forward rate set at time t for a loan that begins at

time T and is to be repaid an instant laterm Drift of the instantaneous forward ratessi i-th volatility of the instantaneous forward rates, 1 ≤ i ≤ ddt Instantaneous time period∆t Short time perioddz Vector of Brownian motion increments (d× 1)d Number of Brownian motionsB(t) Value at time t of a money market account with B(0) = 1E[x] Expectation of xstd(x) Standard deviation of xvar(x) Variance of xcov(x,y) Covariance between x and ycorr(x,y) Correlation between x and yζ Stochastic discount factora Drift of the stochastic discount factorbi i-th volatility of the stochastic discount factor, 1 ≤ i ≤ dN Holdings vector

XIV Commonly Used Symbols

N Holdings vector of risky zero-coupon bondsWt Wealth at time tn Number of assetsτ Maximum maturity of zero-coupon bondsP0 Vector of current prices of risky zero-coupon bondsC Covariance matrixε Second factor in HW2 modelρ Correlation between r and ε in HW2 modelθ Mean reversion levelκr Mean reversion speed of short rate in Vasicek and HW2 modelκε Mean reversion speed of ε in HW2 modelσε Volatility of ε in HW2 modelu(W ) Utility of wealth functionγ Risk aversion parameter in CRRA utility functionx Vector of state variables (d× 1)d Number of state variablesα Drift vector of state variablesβ Volatility matrix of state variablesσt Matrix of volatilities of bond price returns (n× d)w Portfolio weights vectorw∗ Optimum portfolio weights vectorJ Optimal value functionJx Partial derivative of J with respect to xI Identity matrix

1

Introduction

The tools of modern portfolio theory1 are in general use in the equity markets,either in the form of portfolio optimization software or as an accepted frame-work in which the asset managers think about stock selection.2 In the fixedincome market on the other hand, these tools seem irrelevant or inapplicable.Bond portfolios are nowadays mainly managed by a comparison of portfoliorisk measures3 vis a vis a benchmark.4 The portfolio manager’s views aboutthe future evolution of the term structure of interest rates translate them-selves directly into a positioning relative to his benchmark, taking the risksof these deviations from the benchmark into account only in a very crudefashion, i.e. without really quantifying them probabilistically.5 This is quitesurprising since sophisticated models for the evolution of interest rates arecommonly used for interest rate derivatives pricing and the derivation of fixedincome risk measures.6

Wilhelm (1992) explains the absence of modern portfolio tools in the fixedincome markets with two factors:7 historically relatively stable interest ratesand systematic differences between stocks and bonds that make an applicationof modern portfolio theory difficult. These systematic differences relate mainlyto the fixed maturity of bonds. Whereas possible future stock prices becomemore dispersed as the time horizon widens, the bond price at maturity isfixed.8 This implies that the probabilistic models for stocks and bonds have

1 Starting with the seminal work of Markowitz (1952).2 See e.g. Grinold/Kahn (2000), Litterman (2003) or Elton et al. (2003).3 These are commonly partial duration like risk measures, such as level, slope and

curvature durations introduced in Willner (1996).4 See the standard literature on bond portfolio selection, e.g. Fabozzi (2001).5 Usually the tracking error is calculated but doesn’t receive much attention.6 For example to derive a duration measure for swaptions. For a review of commonly

used bond portfolio risk measures see Golub/Tilman (2000).7 Wilhelm (1992), p. 210.8 It must be equal to the face value plus coupon. The fixed income terminology will

be presented in Chapter 2.

2 1 Introduction

to be different and the tools of modern portfolio theory have to be adaptedto be applicable to fixed income instruments.9

In this thesis, we analyze how modern portfolio theory10 and dynamic termstructure models can be used for government bond portfolio optimization.We study the necessary adjustments, examine the models with regard to theplausibility of their results and compare the outcomes to portfolio selectiontechniques used by practitioners.

The question of how to adapt the mean-variance model for bond portfolioselection purposes is hardly new. The earliest attempt is due to Cheng (1962).He modeled the trade-off between rolling over short-term investments and in-vesting at a given spot rate until the investment horizon.11 Hence, he confinedhis analysis to the impact of reinvestment risks on bond portfolio selection.12

The model needs as input probability beliefs about future term structures ofinterest rates (as reinvestment rates). Cheng (1962) suggests analyzing em-pirical data on interest rate movements for a formulation of these probabilitybeliefs.13 He allows for all possible reinvestment tactics until the investmenthorizon.14 One problem is therefore the enormous number of possible tacticsto be considered.15

Bradley/Crane (1972) propose a dynamic bond portfolio selection formu-lation. They argue that the approach by Cheng (1962) doesn’t take into con-sideration the dynamic nature of any portfolio selection problem. But as theyadmit, formulation of a realistic bond portfolio selection problem in theirframework results in extensive data requirements.16 This stems from the factthat one needs to specify an event sequence (possible term structure move-ments) over time with the associated probability beliefs.17 With hindsight, theusage of dynamic term structure models for the generation of these probabilitybeliefs seems obvious. But the first dynamic term structure model was onlyintroduced in 1977 by Vasicek.18 In 1980, Brennan and Schwartz expressed thehope, that using term structure models for bond portfolio selection purposes,will help to reduce the data requirement burden.19

9 Hence the geometric Brownian motion model used to model stocks can’t be usedfor bond price modeling purposes.

10 The static mean-variance model and dynamic continuous-time variants.11 See Cheng (1962), p. 492.12 He ruled out selling bonds before maturity, so market risks or practical strategies

such as rolling down the yield curve (to be introduced in Chapter 4) are ignored.13 See Cheng (1962), p. 491.14 If for example the investment horizon is 10 years, then one possible tactic is

investing for one period at spot interest rate R(0, 1), then for four periods atR(1, 5), then for another two periods at R(5, 7) and at last for three periods atR(7, 10) until the investment horizon.

15 See Cheng (1962), p. 499.16 See Bradley/Crane (1972), p. 150.17 See Bradley/Crane (1972), p. 150.18 Vasicek (1977).

1 Introduction 3

Wilhelm (1992) introduces a bond portfolio selection model using dynamicterm structure models. He identifies the portfolio selection problem as a dy-namic problem but then comes to the conclusion that the problem in suchgenerality is practically unsolvable and hence he confines his analysis to astatic framework.20 He then derives optimum portfolios in a mean-varianceframework where the bond market is governed by the term structure modelof Cox/Ingersoll/Ross (1985). A critical assumption of the model is the rein-vestment of cash flows occurring before the investment horizon at the currentspot interest rate until the investment horizon.21

Fabozzi/Fong (1994) describe the possibility of using portfolio optimiza-tion in the fixed income markets and identify the calculation of the covariancematrix of bond returns as the main obstacle.22 They conclude that “if a co-variance matrix could be created, the bond optimization process could parallelthe analysis for stocks”.23

Elton et al. (2003) also propagate using modern portfolio theory for bondportfolio selection purposes.24 They assume implicitly a short-term investmenthorizon, since they don’t make any assumption regarding the reinvestment ofcash flows. Furthermore, they use separate models for deriving the probabilitybeliefs about expected returns and covariances of returns. For estimation ofexpected returns, they suggest using any of the classical term structure theo-ries25, e.g. the expectation theory.26 The covariance matrix shall be estimatedusing a single- or multi-index model.27 Hence, according to this approach, eachbond is influenced by market risk factors and an idiosyncratic risk factor.28

A recent contribution to this line of research is due to Korn/Koziol (2006).They analyze the problem of an investor who can invest in zero-coupon bondsof different maturities. The underlying bond market uncertainty is drivenby an (multi-factor) affine term structure model.29 The key contribution oftheir paper is an examination of the historical performance of mean-varianceefficient bond portfolios in the German government bond market.

Another line of research analyzes the adaption of continuous-time portfolioselection model based on Merton (1971) to the selection of bond portfolios.

19 See Brennan/Schwartz (1980), p. 406.20 See Wilhelm (1992), p. 216.21 See Wilhelm (1992), p. 216.22 See Fabozzi/Fong (1994), p. 154.23 Fabozzi/Fong (1994), p. 154.24 See Elton et al. (2003), pp. 540–546.25 To be introduced in Chapter 2.5.26 See Elton et al. (2003), p. 540.27 See Elton et al. (2003), pp. 543–546.28 For default-free government bonds, this seems problematic. All government bond

prices (at least of a specific country) are influenced by the same market factors,i.e. interest rates of different maturities. In our opinion there is no room foridiosyncratic risk factors.

29 See Korn/Koziol (2006), p. 4.

4 1 Introduction

Sørensen (1999) examines the problem of an investor with constant relativerisk aversion (CRRA) who can invest into a stock index, a zero-coupon bondand a money market account.30 The term structure of interest rates is gov-erned by the Vasicek (1977) model. He derives a solution using the martingaleapproach by Cox/Huang (1989).

Korn/Kraft (2002) study a pure bond portfolio selection problem where in-terest rates follow either the Vasicek (1977) or the Cox/Ingersoll/Ross (1985)term structure model. They solve the problem using the stochastic control ap-proach. Kraft (2004) extends this analysis by covering additionally the termstructure models by Dothan (1978), Ho/Lee (1986) and Black/Karasinksi(1991).

Munk/Sørensen (2004) solve a pure bond portfolio selection problem ina general Heath/Jarrow/Morton (1992) term structure framework using themartingale approach.

In this thesis we derive the Heath/Jarrow/Morton (1992) term structureframework and two special cases (the Vasicek (1977) and the Hull/White(1994) models) using the stochastic discount factor pricing methodology.Hence we use a different approach than is usually taken.31 Furthermore, weextend the mean-variance bond portfolio selection model proposed by Wil-helm (1992) to the term structure models of Vasicek (1977) and Hull/White(1994). We compare the resulting portfolios to traditional active and passivebond portfolio selection methods. In addition, we derive an explicit analyticsolution to a continuous-time bond portfolio selection model where the bondmarket is driven by the Hull/White (1994) model. As an extension to thecontinuous-time model, we show how foreign currency bonds can be incorpo-rated in the analysis. We introduce an international bond portfolio selectionproblem and give an analytic solution for a simple two-country case.

This thesis is organized as follows. Chapter 2 introduces interest rate ter-minology and the term structure of interest rates. In Chapter 3 we presentthe derivation of the general term structure modeling framework by Heath/Jarrow/Morton (1992) using the stochastic discount factor methodology. Fur-thermore, we derive the Vasicek (1977) and the Hull/White (1994) modelsas special cases. Chapter 4 deals with static mean-variance bond portfolioselection as introduced by Wilhelm (1992). Chapter 5 examines continuous-time bond portfolio selection problems. We derive the solution to a pure bondportfolio selection problem with the Vasicek (1977) and the Hull/White (1994)models using the stochastic control approach. Furthermore, we analyze an in-ternational bond portfolio selection problem and derive an explicit solutionfor a two-country case. Chapter 6 concludes the thesis.

30 See Sørensen (1999), p. 517.31 Originally term structure models were derived using the martingale or the PDE

approach.

2

Bond Market Terminology

By a fixed-income market, we mean that particular sector of the financialmarket on which interest rate sensitive instruments trade.32 In this thesiswe are only interested in the government bond market, i.e. that part of thefixed-income market where government bonds are traded.

2.1 Characteristics of Bonds

A bond represents a claim on a prescribed sequence of payments.33 The issuerof the bond (the borrower) is committed to paying back to the bondholder(the lender) the cash amount borrowed plus periodic interest payments.34

One important characteristic of a bond is the nature of its issuer.35 Typi-cal bond issuers are (municipal and federal) governments and (domestic andforeign) corporations. Government bonds are assumed to be default-free, i.e.the above mentioned sequence of payments is ex ante known with certainty.

Another key feature of a bond is the maturity (date). This is the date onwhich the debt will cease to exist36 and the borrower will redeem the issue bypaying the face value.37 By term to maturity we mean the time to maturity.Face value (or principal, or par value) is the amount the issuer agrees to repayat maturity38 or the nominal amount borrowed by the debtor.

Regarding the sequence of payments the bondholder is entitled to, onecan distinguish between two main types of bonds. Zero-coupon bonds (or zerobonds, zeroes or discount bonds) pay only the principal at maturity. No otherpayments are made during the life of the bond. Coupon bonds however promise32 See Musiela/Rutkowski (1997), p. 265.33 See Shiller (1990), p. 633.34 See Martellini/Priaulet/Priaulet (2003), p. 3.35 See Fabozzi (2000), p. 3.36 See Focardi/Fabozzi (2004), p. 51.37 See Fabozzi (2000), p. 4.38 See Focardi/Fabozzi (2004), p. 52.

6 2 Bond Market Terminology

a stream of payments. Like a zero-coupon bond, they pay the principal at thematurity date, but the bondholder also receives interest payments at regularintervals. These intermittent payments are called coupons. The amount ofthe coupon payments is determined by multiplying the coupon rate by theprincipal.39 Interest payments are usually made annually or semiannually.

Bond prices are quoted in two different forms.40 The dirty price is theactual amount in return for the right to the full amount of each future couponpayment and the redemption proceeds.

The clean price is an artificial price which is, however, the most-quotedprice in the marketplace.41 It is equal to the dirty price minus accrued interest.The accrued interest is equal to the amount of the next coupon paymentmultiplied by the proportion of the current inter-coupon period so far elapsed,i.e. the buyer of the bond “compensates” the seller and pays him that partof the current coupon he already “earned”. The popularity of the clean pricerelies on the fact that it does not fall as a result of a coupon payment.42

For further information regarding bond market conventions (day countconventions etc.) see Brown (1994), Stigum/Robinson (1996) and Krgin(2002).

2.2 Interest Rates

The spot interest rate R(t, T ) designates the rate of interest per period (usuallya year) charged on a loan that begins at time t and is paid back at time T ,with t ≤ T .43 Spot interest rates can be derived from the prices of zero-coupon bonds. By convention we set the zero-coupon bond’s principal to 1unit of account. Let P (t, T ) be the price44 of a zero-coupon bond at time twith maturity T , then there is the following (defining) relationship betweenzero-coupon bond prices and interest rates45

P (t, T ) = exp (−(T − t)R(t, T )) (2.1)

We can solve for R(t, T ) and obtain

R(t, T ) = − 1T − t

ln(P (t, T )) (2.2)

39 With an adjustment for the payment frequency.40 See Cairns (2004), p. 2.41 See Cairns (2004), p. 3.42 See Tuckman (2002), p. 55.43 The spot interest rate is normally denoted by r(t, T ) but we reserve r for the

short rate of interest.44 For zero-coupon bonds the clean and dirty prices are equal since there are no

intermediate payments and hence no accrued interest.45 We assume continuous compounding.

2.2 Interest Rates 7

In later chapters we concentrate on modeling one specific spot interest rate:the short rate. The short rate is the spot interest rate with instantaneousmaturity, i.e.46

r(t) ≡ limT→t

R(t, T ) (2.3)

The short rate is – of course – unobservable in the market. Therefore a suitableproxy for the short rate has to be found in any empirical work.

The forward interest rate tR(T, τ) is the interest rate set at time t for aloan that begins at time T and is repaid at time τ , with t ≤ T ≤ τ .47 It isdefined by the following relationship48

exp(R(t, τ)(τ − t)) = exp(R(t, T )(T − t)) exp(tR(T, τ)(τ − T )) (2.4)= exp(R(t, T )(T − t) +t R(T, τ)(τ − T )) (2.5)

We solve for tR(T, τ) and obtain

tR(T, τ) =1

τ − Tln

(P (t, T )P (t, τ)

)(2.6)

which can also be written as49

tR(T, τ) = R(t, τ)(τ − t)(τ − T )

−R(t, T )(T − t)(τ − T )

(2.7)

Another group of interest rate models to be introduced in a later chapter50,focuses on a particular set of forward rates namely the instantaneous forwardrates f(t, T ). These are interest rates set at time t for loans that begin attime T and will be repaid an instant later (at time T +dt). They are the basicbuilding blocks of the fixed income market since every interest rate can beexpressed in terms of instantaneous forward rates. Mathematically they aredefined as follows

f(t, T ) ≡ limτ→T

tR(T, τ) (2.8)

With tR(T, τ) from (2.6) we obtain the following relationship51

f(t, T ) = limτ→T

1τ − T

ln(

P (t, T )P (t, τ)

)

= − ∂

∂Tln P (t, T )

= − 1P (t, T )

∂P (t, T )∂T

(2.9)

46 See Cairns (2004), p. 6.47 See Martellini/Priaulet/Priaulet (2003), p. 52.48 This relation must hold in an arbitrage-free market. Otherwise one could arbitrage

by taking positions in the spot and forward market.49 Use equation (2.2).50 The Heath/Jarrow/Morton (1992) framework.51 See Cairns (2004), p. 5. Note further that f(t, t) = r(t).


Equation (2.9) resembles the modified duration formula for bonds.52 Itcan be thought of as the percentage gain in the price of a zero-coupon bondwith maturity T if the maturity was to be extended by a small time period.

Zero-coupon bond prices at time t for different maturities T can be calcu-lated from the instantaneous forward rate curve at time t53

P (t, T ) = exp

(−

∫ T

t

f(t, u) du

)(2.10)

The yield to maturity yT of a bond maturing at time T is the internal rateof return of that bond, i.e. the interest rate that when used for discountingthe future payments results in a net present value of zero. Or in other words,discounting all future payments with the yield to maturity gives the dirtyprice P of this coupon bond with maturity T , coupon rate c and face valueF .

P =T∑

τ=1

cF

(1 + yT )τ+

F

(1 + yT )T(2.11)

It should be simply regarded as another way of quoting the price of thatparticular bond. The yield to maturity of a zero-coupon bond maturing attime T equals the spot interest rate for that maturity. For coupon bonds, theyield to maturity is not the actual holding period return of the bond, becausethis holding period return depends on the reinvestment rates of the coupons.Only if all coupons could be reinvested at the yield to maturity, the holdingperiod return would equal the yield to maturity. The yield to maturity isbond specific and does not give detailed information about the interest ratesfor specific dates, since it is only a complex average of the respective spotinterest rates. The collection of the yield to maturity of all quoted bonds –the yield curve – gives only a crude indication of the interest rates on themarket since yields are bond and maturity specific. A better construct (atleast for academic purposes) is the term structure of interest rates.

2.3 Term Structure of Interest Rates

The term structure of interest rates at a given time is the functional rela-tionship between spot interest rates and term to maturity.54 It can also bedescribed by forward rates or discount factors, since they contain the sameinformation.55

For the term structure to be observable in the market, zero-coupon bondsof different maturities must trade. In reality, only zero-coupon bonds of veryfew maturities (typically smaller than a year) trade.56

52 Modified duration is defined as MD = − 1P (r)

∂P (r)∂r

.53 Solve equation (2.9) for P (t, T ).54 See Wilhelm (1995), p. 2052.55 See Martellini/Priaulet/Priaulet (2003), p. 63.

2.4 Estimating the Term Structure of Interest Rates 9

But an abundance of traded coupon bonds exist in the fixed income mar-kets. Coupon bonds can be thought of as portfolios of zero-coupon bonds.Hence, only specific portfolios of zero-coupon bonds trade.

The problem is then to extract individual zero-coupon prices from theprices of particular portfolios of zero-coupon bonds. From these (theoretical)zero-coupon bond prices the corresponding spot rates can then be calculated.

When the term structure of interest rates is known, one can hence valueall assets with known future cash flows, e.g. government bonds.

2.4 Estimating the Term Structure of Interest Rates

In this section we give only a brief overview of term structure estimation tech-niques since there exists a rich literature on the subject and a full introductionwould be outside the scope of this thesis.

Under some very restrictive assumptions on the nature of the tradedcoupon bonds, the term structure of interest rates can be calculated by thebootstrap method.57 This method uses an iterative process. One starts with aone-period zero-coupon bond58 and calculates the spot interest rate R(0, 1).This interest rate is then used to calculate the spot interest rate R(0, 2) giventhe price of a two-year coupon bond. This method works only if the couponbonds have the same coupon payment dates.59 Furthermore, it produces nota smooth term structure but only a collection of interest rates for specificmaturities. It is common practice to interpolate the interest rates betweentwo points linearly.60 These restrictive assumptions are usually not met in thereal world, but nevertheless the bootstrap method seems to be quite popularamong practitioners.61

A further approach is to assume a specific (parameterized) functional formfor the term structure (or the discount function).62 The objective is then tochoose the parameters in such a way as to fit the model bond prices as closelyas possible to observed bond prices. Two frequently applied parametrizationtechniques are the cubic spline method and the Nelson-Siegel approach.63

56 Recently some governments introduced stripping, i.e. the creation of zero-couponbonds out of coupon bonds. But the coupon-only and principal-only strips are notas liquid as the coupon bonds and so their prices should not be used to constructthe term structure of interest rates, because of the liquidity effect.

57 See Caks (1977), p. 104.58 Every coupon bond becomes a zero-coupon bond eventually.59 See Munk (2004b), p. 22.60 See Hull (2005), p. 84.61 Bootstrapping is also used for the derivation of the swap curve. It is common to

use money market rates, futures and swap rates for its construction.62 With some additional assumptions one therefore obtains a smooth term structure.63 See Munk (2004b), p. 21.


The cubic spline method was first introduced by McCulloch (1971). Thismethod assumes that the maturity axis is divided into subintervals and thatseparate functions of the same type (so-called splines) are used to describethe discount function in the different subintervals.64 The problem with thismethod is that the derived forward rate curve will usually be quite ruggedand that the curve is quite sensitive to the bond prices and the location ofthe knot points.65

Nelson/Siegel (1987) propose a parsimonious model of the term structurethat has become quite popular particularly among central banks. The modelwas later extended by Svensson (1994). Their parametrization of the instanta-neous forward rate curve allows for quite flexible shapes of the term structureof interest rates although it relies on only 4 parameters. Furthermore theforward rate curve is smooth.66

For further information about term structure estimation techniques seeChapter 4 in Martellini/Priaulet/Priaulet (2003), Chapters 10 and 11 inChoudhry (2004) and the book by Anderson et al. (1996).

2.5 Classical Theories of the Term Structure of InterestRates

After the estimation of the term structure the natural question arises of whatdetermines its shape.67 There exist several yield curve theories that try toanswer this question. We only give a short overview here since we will adopta different approach in this thesis. It is intended to highlight the differencesbetween the classical theories and modern arbitrage-free theories of the term-structure of interest rates.

The expectations theory was developed by Hicks (1939) and Lutz (1940). Ithas a number of variants all relying on the idea that current interest rates arelinked to expected future rates. For a critical examination see Cox/Ingersoll/Ross (1981).

The liquidity preference theory introduced by Hicks (1939) finds that theexpectations theory ignores the investor’s risk aversion and argues that ex-pected returns on bonds with longer maturities should be higher than forshorter bonds to compensate for the higher price fluctuation of longer bonds.68

According to this hypothesis, the term structure of interest rates must be nor-mal, i.e. spot rates rise with longer maturity.

In contrast, the market segmentation theory by Culbertson (1957) claimsthat investors want to invest in an appropriate set of bonds and maturity64 See Munk (2004b), p. 25.65 See Munk (2004b), p. 29.66 This is due to the fact, that the functional form of the forward rate curve is an

input (assumption) of the model unlike in the cubic spline model.67 Traditionally one distinguishes normal, flat and inverse term structure shapes.68 See Munk (2004b), p. 117.

2.7 Empirical Properties of the Term Structure of Interest Rates 11

segments that are suitable for their purpose.69 Since different investors actin different ways there is no reason to assume that the movement of interestrates in different maturity segments is interrelated.

A more realistic version of this hypothesis is the preferred habitat theoryintroduced by Modigliani/Sutch (1966). They argue that although each in-vestor might prefer to invest in a particular set of bonds, he should be willingto invest in different bonds if he is sufficiently compensated for doing so.70

The different segments of the bond market are therefore not completely inde-pendent of each other.

2.6 Arbitrage-Free Term Structure Theories

In this thesis, the classical theories are not considered because of their short-comings regarding the pricing of bonds. We consider arbitrage free pricingtheories which pull together the classical theories in a mathematical preciseway.71 These theories will be presented in Chapter 3.

2.7 Empirical Properties of the Term Structure ofInterest Rates

Before we turn to the topic of (arbitrage-free) interest rate modeling, we needto take a closer look at some empirical properties of the term structure. Arealistic model should at least possess some of these properties. Rebonato(1998) describes some features of desirable interest rate models.72 Interestrates should not become negative and display mean reversion.73 Mean rever-sion refers to a level-dependent drift of a stochastic process, i.e. the drift ispositive (negative) when the current value of the stochastic process is below(above) a certain level.74 The correlation between interest rates of differentmaturities generated by the model should be positive but imperfect.75 Theterm structure of volatility is normally not flat but humped.76 In a humpedvolatility structure, the volatility first rises and then declines.

As usual, model choice is a trade-off between realism and (analytical ornumerical) tractability. In this thesis, we focus more on tractability. In Chapter3, we will present the general framework for term structure modeling andderive two analytically tractable models as special cases.69 See Cairns (2004), p. 12.70 See Munk (2004b), p. 118.71 See Cairns (2004), p. 13.72 See Rebonato (1998), pp. 233–234.73 See Rebonato (1998), p. 233.74 The stochastic process for the short rate in the Vasicek term structure model to

be introduced in Chapter 3 possesses this feature.75 See Golub/Tilman (2000), p. 89.76 See Golub/Tilman (2000), p. 89.

3

Term Structure Modeling in Continuous Time

3.1 Introduction

The main inputs of portfolio selection models are the expected values andcovariances of the assets under consideration. In equity portfolio selection,the expected values and covariances are oftentimes estimated by analyzingthe historical time series of the stocks. Because of bond characteristics andproperties of bond portfolio selection models that will be discussed in greaterdetail in Chapter 4, such an approach is generally ruled out for fixed incomeinstruments. In order to determine the bond portfolio selection parametersconsistently, a theoretical model for the evolution of bond prices over time isneeded.77

We introduced the term structure of interest rates in Chapter 2. Giventhe term structure at time t, the time t value of all bonds with known futurepayoffs can be calculated.78 Furthermore, future term structures determinefuture bond prices. Bonds can therefore be regarded as term structure (orinterest rate) derivatives since their value depends on interest rates. Hence, atheory about the dynamics of the term structure yields at the same time amodel for the movements of bonds over time.79

Dynamic term structure modeling – i.e. modeling the evolution of the termstructure of interest rates over time in an arbitrage-free fashion – is one of themost heavily researched areas in financial economics.80

An early attempt to model the term structure dynamics was the durationmodel.81 It assumes that the term structure of interest rates is flat and movesonly in a parallel fashion. On first thought this seems to be a reasonableassumption, but it turns out that this is problematic for two reasons: (i)

77 See Wilhelm (1992), p. 213.78 See Wilhelm (1992), p. 209.79 See Wilhelm (1992), p. 209.80 Choudhry (2004), p. 178.81 The model dates back to the seminal analysis of Macaulay (1938).

14 3 Term Structure Modeling in Continuous Time

such term structure movements are empirically improbable and (ii) if suchmovements would occur, it would present an arbitrage opportunity.82 Thesecond argument is of course much more serious from a theoretical point ofview.

Using the duration model for bond portfolio management purposes istherefore unreasonable. Due to the presence of arbitrage opportunities, therewould not exist an optimal solution to the portfolio selection problem, in thesense that every possible portfolio would be dominated. Consequently – atleast for bond portfolio selection purposes – there is a need for better interestrate models.83

It is generally accepted that term structure modeling theory started withthe seminal paper of Vasicek (1977). Since then a vast number of interest ratemodels have been proposed by academics and practitioners alike.

Section 3.2 gives an overview of possible term structure modeling ap-proaches. In Section 3.3 we introduce the Heath/Jarrow/Morton (1992) frame-work and derive a general zero-coupon bond pricing equation using stochasticdiscount factors. Sections 3.4 and 3.5 derive two special cases of the generalHeath/Jarrow/Morton (1992) framework, that will be used in later chapterson bond portfolio optimization.

3.2 Interest Rate Modeling Approaches

There are several approaches to modeling the term structure of interest rates.A possible classification distinguishes between whole yield curve models andshort rate models.84

Whole Yield Curve Models

The most general approach is to specify the stochastic processes followed byinstantaneous forward rates f(t, T ).85 For a fixed maturity T , we can writethis process as

df(t, T ) = m(t, T )dt + s1(t, T )dz1

where m(t, T ) is the drift, s1(t, T ) is the instantaneous standard deviation anddz1 is a Brownian motion. This approach has been proposed in a groundbreak-ing paper by Heath/Jarrow/Morton (1992) (hereafter HJM). HJM model thewhole term structure of instantaneous forward rates, i.e. an infinite numberof interest rates. They show that the drift of the forward rates must follow

82 See Cairns (2004), p. 18.83 See Baz/Chacko (2004), p. 108.84 For ease of exposition we assume in this section that a single Brownian motion

drives the whole term structure.85 Instantaneous forward rates have been introduced in Equation (2.9).

3.2 Interest Rate Modeling Approaches 15

from the specification of the volatilities and the market prices of interest raterisk.86

Hull/White (1996) show that any model of zero-coupon bond prices canbe converted into an equivalent model of instantaneous forward rates and viceversa.87 The dynamics of the zero-coupon bond prices can be written as88

dP (t, T )P (t, T )

= µ(t, T )dt + σ1(t, T )dz1

where P (t, T ) is the time t price of a zero-coupon bond with maturity T ,µ(t, T ) is the drift and σ1(t, T ) is the bond price volatility. It has been shownthat in an arbitrage-free setting, the drift of the bond prices is a particularfunction of the short rate r, the volatilities σ1 and the market prices of interestrate risk. The volatility of the bond price must be a function of maturity sinceit must decline as maturity approaches. Since the price of the bond at maturityis known with certainty, we must have that the volatility at maturity is zero,i.e. σ1(T, T ) = 0.89

The advantage of specifying a model in terms of the processes followed byeither bond prices or forward rates is that the model is automatically consis-tent with the initial term structure, since this initial term structure determinesthe initial values of the variables being modeled.90 The disadvantage is thatthe model is usually non-Markov91 and therefore slow computationally.92 Ashas been shown, a forward rate process is structurally simpler than a bondprice process in that it usually does not depend on the short rate r.93

Models of the Short Rate

The most widely used approach is to model the evolution of one instanta-neous forward rate only, namely the short rate r(t).94 The economic intuitionbehind short rate models arises from the observation that bond prices tend tomove together, i.e. changes in bond prices are highly correlated.95 Short rate86 See Chapter 3.3.4 for a derivation of this drift condition.87 Hull/White (1996), pp. 261–262.88 See Hull/White (1996), p. 228.89 See Hull/White (1996), p. 229.90 See Hull/White (1996), p. 229.91 A stochastic process has the Markov property if the conditional distribution of

its future values depends only on the current value and not on the past. SeeCvitanic/Zapatero (2004), p. 65.

92 See Hull/White (1996), p. 229.93 Hull/White (1996), p. 229.94 It has been common to assume that only one factor drives the term structure.

According to Martellini/Priaulet/Priaulet (2003), p. 388, there is a general con-sensus among researchers that this factor should be the short rate.

95 See Table 3.1 on page 70 of Martellini/Priaulet/Priaulet (2003) for empiricalresults of interest-rate co-movements.


changes are thus used as a proxy for changes in the level of the whole termstructure.96 Empirical studies – for example Litterman/Scheinkman (1991)– confirmed that level changes account for a large part of the dynamics ofthe term structure.97 Short rate models propose a diffusion process for theevolution of the short rate98

dr(t) = α(r, t)dt + σr(r, t)dz1 (3.1)

where α(r, t) is the drift of the short rate and σr(r, t) is its volatility. Thedynamics of other interest rates is derived endogenously. Since α and σr onlydepend on the value of the short rate, the model is always Markov.99 Thismodel is automatically consistent with the current value of the short ratebut not necessarily with other interest rates.100 In other words, because onlythe dynamics of the short rate is specified, the only exogenously given assetis the money market account101 and zero-coupon bond prices are consideredderivatives on the short rate.102

Whole Yield Curve Models versus Short Rate Models

Whole yield curve models can be differentiated from the short rate modelswith regard to various criteria103

• Traded assets. The only traded asset in the short rate models is the moneymarket account, zero-coupon bonds are regarded as derivatives on the shortrate. The HJM framework assumes that zero-coupon bonds of all matu-rities trade (including one with instantaneous maturity, i.e. the moneymarket account).

• Completeness. A market is said to be complete if any contingent claimcan be replicated with existing securities.104 Short rate models are notcomplete. , because there is (at least) one source of risk and just one tradedasset (the money market account). HJM models are complete, becausethere are d risk sources and an infinite number of traded assets.

• Freedom of arbitrage. Short rate models are arbitrage-free per construc-tion, because the only traded asset (and therefore the only possible port-folio) is the money market account. Furthermore since holding of cash isprohibited, negative interest rates don’t lead to arbitrage opportunities.

96 See Martellini/Priaulet/Priaulet (2003), p. 388.97 See Martellini/Priaulet/Priaulet (2003), p. 388.98 Svoboda (2004), p. vii.99 See Hull/White (1996), p. 229.

100 See Hull/White (1996), p. 229.101 The money market account or cash account yields the current short rate, see

Chapter 3.3.2.102 See Bjork (1998), p. 242.103 See Branger/Schlag (2004), p. 160.104 See Cvitanic/Zapatero (2004), p. 88.

3.3 Heath/Jarrow/Morton (1992) 17

The HJM framework is only free of arbitrage opportunities if the driftcondition105 is met.

• Matching of the initial term structure. Time-homogeneous short rate mod-els106 cannot generally fit the current term structure because a finite num-ber of parameters cannot be chosen in such a way as to fit an infinitenumber of prices (or interest rates). Time-inhomogeneous short rate mod-els solve this problem by making some (or all) parameters time-dependent.The HJM model fits the current yield by construction because it is takenas an input.

• Market prices of risk. In a short rate model, the drift and volatility ofthe short rate and market price of risk must be specified. It is particularlydifficult to estimate the drift of the short rate. In a HJM model the drift ofthe interest rates follow from arbitrage considerations and therefore onlythe current term structure, the volatilities of the forward rates and themarket prices of risk must be known.

In the following sections we present the general HJM framework and derivetwo well-known term structure models as special cases, the Vasicek (1977)model and the Hull/White (1994) two-factor model.

3.3 Heath/Jarrow/Morton (1992)

3.3.1 Introduction

The general framework for arbitrage-free interest rate modeling in continuoustime was proposed in a pioneering paper by HJM. The HJM framework is amulti-factor model, i.e. the term structure of interest is subject to multipleshocks. It can be shown that all previously developed107 interest rate modelsare special cases of this framework.108

HJM start with the observed term structure of interest rates in its instan-taneous forward rate form.109 For a fixed but arbitrary maturity T ∈ [t, τ ],f(t, T ) satisfies110

df(t, T ) = m(t, T )dt +d∑

i=1

si(t, T )dzi(t) (3.2)

where z1, . . . , zd are d independent standard Brownian motions, m(t, T ) is thedrift and si(t, T ) is the i-th volatility of f(t, T ). In its most general form, both

105 To be derived in Chapter 3.3.4.106 If both the drift and the volatility are independent of time, the diffusion is said

to be time-homogeneous, otherwise it is said to be time-inhomogeneous.107 This includes all short rate models.108 Svoboda (2004), p. 124.109 See Branger/Schlag (2004), p. 126.110 Heath/Jarrow/Morton (1992), p. 80.


the drift and the volatility may depend on the entire forward rate curve attime t.111

Consequently, the whole term structure of interest rates – an infinite num-ber of stochastic processes – is modeled. But since the model has only finiteBrownian motions as risk sources, there must be some kind of arbitrage-freecondition for the forward rate drift (the HJM drift condition).112 This driftcondition ensures that the traded zero-coupon bonds form an arbitrage-freemarket.113

This section is organized as follows. Section 3.3.2 derives the dynamicsof traded zero-coupon bond prices that must follow from the specification ofthe instantaneous forward rates in Equation (3.2). In Section 3.3.3 we showhow the prices of interest rate derivatives can be obtained by employing thestochastic discount factor approach. In Section 3.3.4 we derive the HJM driftcondition and Section 3.3.5 derives the short rate dynamics that provides aformal link to the special cases (short rate models) introduced later in thechapter.

3.3.2 Dynamics of Traded Securities

In the HJM framework, it is assumed that a money market account B(t) andzero-coupon bonds P (t, T ) of different maturities T trade.114

The dynamics of the money market account

B(t) = B(0) exp(∫ t

0

f(u, u)du

), B(0) = 1

can be obtained by a straightforward application of Ito’s lemma

dB(t)B(t)

= f(t, t)dt (3.3)

With Equation (2.10) we can derive zero-coupon bond prices from the instan-taneous forward rate curve

P (t, T ) = exp

(−

∫ T

t

f(t, u)du

)(3.4)

Applying the stochastic Fubini theorem115 and Ito’s lemma, we obtain thedynamics of zero-coupon bonds116

111 See Heath/Jarrow/Morton (1992), p. 80. For notational convenience this depen-dence is not indicated in Equation (3.2).

112 See Branger/Schlag (2004), p. 126.113 We derive the drift condition in Chapter 3.3.4.114 See Heath/Jarrow/Morton (1992), p. 79.115 See Heath/Jarrow/Morton (1992), p. 99.116 For a derivation see Appendix A.1.


dP (t, T )P (t, T )

= µ(t, T )dt−d∑

i=1

σi(t, T )dzi(t) (3.5)

where

µ(t, T ) = f(t, t)−∫ T

t

m(t, u)du +12

d∑

i=1

(∫ T

t

si(t, u)du

)2

(3.6)

σi(t, T ) =∫ T

t

si(t, u)du (3.7)

3.3.3 Arbitrage-Free Pricing

The prices of interest rate derivative securities in an arbitrage-free market canbe obtained by multiple techniques. Originally, the derivation of the interestrate models followed either the PDE approach or the martingale approach.117

In this thesis, we use the stochastic discount factor methodology for pricinginterest-rate derivative securities.118

The fundamental theorem of asset pricing states that the time t price P (t)of a financial claim can be obtained in an arbitrage-free market by taking theexpectation with respect to the real-world probabilities over the product ofstochastic discount factor ζ and payoff P (T )119

P (t) = Et

[ζT

ζtP (T )

](3.8)

We now assume the following dynamics for the stochastic discount factor120

dζ(t) = a(t)ζ(t)dt + ζ(t)d∑

i=1

bi(t)dzi(t)

where a(t) is the drift and bi(t) is the volatility of the stochastic discountfactor with regard to the i-th Brownian motion.

For the fixed income market to be free of arbitrage, the drift and volatilitiesof the stochastic discount factor cannot be chosen arbitrarily. Jin/Glasserman

117 For a brief exposition of the two approaches see Cairns (2004), p. 55 andCairns (2004), p. 60.

118 For a book-length treatment of asset pricing that contains a chapter on thestochatic discount factor methodology see Cochrane (2005). Wilhelm (2005) de-rives Gaussian interest rate models from the specification of the stochastic dis-count factor. The Ho-Lee interest rate model was derived by stochastic discount-ing already in Wilhelm (1999). For stochastic discounting in a discrete time settingsee Wilhelm (1996).

119 If the market is free of arbitrage and complete, there exists a unique stochasticdiscount factor, see Harrison/Kreps (1979) and Harrison/Pliska (1981).

120 Based on the one-factor formulation in Baz/Chacko (2004), p. 51.


(2001) show that the drift and the volatilities of the stochastic discount factormust be related to the short rate f(t, t) and the market prices of interest raterisk λi(t)121 in the following way122

a(t) = −f(t, t) (3.9)bi(t) = λi(t) ∀ i = 1, . . . , d (3.10)

The drift of the stochastic discount factor must be equal to the negative shortrate and the volatilities must be equal to the market prices of interest raterisk. The dynamics of the stochastic discount factor can then be given as

dζ(t) = −f(t, t)ζ(t)dt + ζ(t)d∑

i=1

λi(t)dzi(t) (3.11)

Equation (3.11) can be solved explicitly.123 We obtain

ζ(T ) = ζ(t) exp

(∫ T

t

−f(u, u)du +d∑

i=1

∫ T

t

λi(u)dzi(u)−d∑

i=1

12

∫ T

t

λi(u)2du

)

and the arbitrage-free prices of zero-coupon bonds can be derived with thefollowing pricing equation using (3.8) and P (T, T ) = 1

P (t, T ) = Et

[exp

(−

∫ T

t

f(u, u)du +d∑

i=1

∫ T

t

λi(u)dzi(u)

−d∑

i=1

12

∫ T

t

λi(u)2du

)](3.12)

The valuation of zero-coupon bonds hence requires a specification of the mar-ket prices of interest rate risk λi and of the short rate evolution. Before wederive an expression for the short rate, we derive the HJM drift condition.

3.3.4 Excursus: The HJM Drift Condition

The groundwork laid in the last sections allows us to derive the classical HJMdrift condition. First, we calculate the deflated price process Y (t, T ) with

Y (t, T ) = ζ(t)P (t, T )

121 The market price of interest rate risk gives the expected return over the risklessrate per unit of volatility. It must be equal for all traded assets, see Branger/Schlag (2004), p. 115.

122 See Jin/Glasserman (2001), p. 195.123 For a derivation see Appendix A.2.


The deflated price process is a martingale.124 An application of Ito’s lemmawith dP (t, T ) from (3.5) and dζ from (3.11) yields the dynamics of Y 125

dY (t, T )Y (t, T )

=(−f(t, t) + µ(t, T )−

d∑

i=1

λi(t)σi(t, T )

)dt +

d∑

i=1

(λi(t)− σi(t, T ))dzi(t)

Since Y (t, T ) is a martingale, the drift of Y (t, T ) must be equal to zero. Weobtain

µ(t, T ) = f(t, t) +d∑

i=1

σi(t, T )λi(t) (3.13)

We insert µ(t, T ) and σi(t, T ) from (3.6) and (3.7) into (3.13) and obtain

∫ T

t

m(t, u)du =12

d∑

i=1

(∫ T

t

si(t, u)du

)2

−d∑

i=1

λi

(∫ T

t

si(t, u)du

)

Differentiating this equation with respect to T yields the so-called HJM driftcondition126

m(t, T ) =d∑

i=1

si(t, T )

((∫ T

t

si(t, u)du

)− λi(t)

)(3.14)

The drift of the forward rates cannot be chosen independently but it resultsfrom the specification of the volatility structure of the forward rates and themarket prices of risk.127

3.3.5 The Short Rate of Interest

The short rate is defined as the instantaneous forward rate with instantaneousmaturity, i.e. it can be obtained as the limit T → t of the instantaneouslimit forward rate function f(t, T ). The integrated dynamics of f(t, T ) can bederived with (3.2) and (3.14). It follows that

f(t, T ) =f(0, T ) +d∑

i=1

∫ t

0

si(u, T )

((∫ T

u

si(u, s)ds

)− λi(u)

)du

+d∑

i=1

∫ t

0

si(u, T )dzi(u)

124 This follows from the definition of the stochastic discount factor, see Duffie (1996),p. 103. A stochastic process X is a martingale if Et(Xs) = Xt, see Duffie (1996),p. 22.

125 For a derivation see Appendix A.3.126 See Heath/Jarrow/Morton (1992), p. 84.127 See Branger/Schlag (2004), p. 129.


We obtain the short rate by letting T → t

f(t, t) =f(0, t) +d∑

i=1

∫ t

0

si(u, t)((∫ t

u

si(u, s)ds)− λi(u))

du

+d∑

i=1

∫ t

0

si(u, t)dzi(u) (3.15)

The time t short rate hence depends on the initial instantaneous forward ratef(0, t), the market prices of interest rate risk and the forward rate volatilities.

3.3.6 Special Cases

Particular interest rate models can be derived by applying the general pricingequation (3.12) with f(t, t) from (3.15) and specifying

• volatility function(s) of the instantaneous forward rates si(t, T )• functional form of the initial instantaneous forward rate curve f(0, t)• the market price(s) of risk λi(t)

In the next two sections, we introduce two special cases of the generalHJM interest rate framework

• One-factor Vasicek (1977) model• Two-factor Hull/White (1994) model

We chose these models because they are quite realistic term structuremodels and – of major importance for our analysis – analytically tractable.The Vasicek model can be obtained from the general HJM framework bysetting128

Table 3.1. Vasicek (1977) model: Special case of HJM.

Item Vasicek (1977) model

s(t, T ) σr exp(−κ(T − t))

λ(t) λ

f(0, T ) θ + exp(−κT )(f(0, 0)− θ) + λσrκ

(1− exp(−κT ))− σ2r

2κ2 (1− exp(−κT ))2

In a similar fashion, the HW2 model can be derived by setting129

128 A derivation of the model follows in the next part of the chapter.129 With g(0, T ) defined in Equation (A.6) in Appendix A.4. A derivation of the

model follows in the next part of the chapter.

3.4 Vasicek (1977) 23

Table 3.2. Hull/White (1994) model: Special case of HJM.

Item Hull/White (1994) model

s1(t, T )e(t−T )κr (κr−κε)σr−(e(t−T )κr−e(t−T )κε)%σε

κr−κε

s2(t, T )(−e(t−T )κr +e(t−T )κε)

√1−%2σε

κr−κε

λ1(t) λ1

λ2(t) λ2

f(0, T ) g(0, T ) + r(0)e−κrT + ε(0) e−κεT−e−κrT

κr−κε

In chapters 3.4 and 3.5 we will derive and analyze these special cases ofthe general HJM interest rate modeling framework.

3.4 Vasicek (1977)

3.4.1 Introduction

In 1977 Vasicek proposed what is known to be the first arbitrage-free dynamicinterest rate model.130 He assumed that the term structure of interest rates iscompletely determined by the current value of only one random variable – theshort rate of interest. The short rate follows an Ornstein-Uhlenbeck process.This is a stationary Markov process with normally distributed increments.131

The behavior of the short rate can be described by the following SDE132

dr(t) = κ(θ − r(t))dt + σrdz1(t) (3.16)

where κ > 0, θ and σr are constant.133 θ designates the mean reversion level,κ is the reversion speed and σr is the volatility of the short rate. An Ornstein-Uhlenbeck process exhibits mean reversion. The drift is positive when r(t) < θand negative when r(t) > θ. The process is therefore pulled towards θ. Themagnitude of the pull depends on the reversion speed κ.

3.4.2 Derivation of Zero-Coupon Bond Prices

In Chapter 3.3.3 we obtained a general pricing equation for zero-couponbonds.134 We now want to specify the initial forward rate curve, volatilitystructure and the market prices of interest rate risk in order to obtain a for-mula for zero-coupon bond prices in the Vasicek (1977) model.

130 Vasicek (1977).131 Vasicek (1977), p. 185.132 Vasicek (1977), p. 185.133 Vasicek (1977), p. 185.134 See Equation (3.12).


The specifications for the Vasicek (1977) model in the HJM framework ofChapter 3.3 are as follows. The model is a one-factor interest rate model, i.e.d = 1. The forward rate volatilities are assumed to be of the following form

s1(t, T ) = s(t, T ) = σr exp(−κ(T − t)) (3.17)

and the market price of interest rate risk is a constant, i.e.

λ1(t) = λ(t) = λ (3.18)

The initial instantaneous forward rate curve is given by

f(0, T ) =θ + exp(−κT ))(f(0, 0)− θ)

+ λσr

κ(1− exp(−κT ))− σ2

r

2κ2(1− exp(−κT ))2 (3.19)

With these specifications, Equation (3.15) determines the short rate in theVasicek model. We obtain135

r(T ) = f(T, T )

= f(t, T ) +∫ T

t

s(u, T )

(∫ T

u

s(u, s)ds− λ

)du +

∫ T

t

s(u, T )dz(u)

= r(t)e−κ(T−t) + θ(1− e−κ(T−t)) + σr

∫ T

t

e−κ(T−u)dz(u) (3.20)

Now, we are in a position to compute zero-coupon bond prices, since thegeneral bond pricing equation (3.12) is completely specified. It follows that

P (t, T ) = Et

[exp

(−r(t)

1− e−κ(T−t)

κ− θ

(−1 + e−κ(T−t) + κ(T − t))κ

−12λ2(T − t) +

σr

κ

∫ T

t

((e−κ(T−s) − 1) + λ)dz(s)

)](3.21)

There is only one stochastic expression in Equation (3.21), hence

P (t, T ) = exp(−r(t)

1− e−κ(T−t)

κ− θ

(−1 + e−κ(T−t) + κ(T − t))κ

−12λ2(T − t)

)Et

exp

(σr

κ

∫ T

t

((e−κ(T−s) − 1) + λ)dz(s)

)

︸︷︷︸x

(3.22)

135 This could be obtained as well by solving (3.16).

3.4 Vasicek (1977) 25

Since x is a normally distributed random variable, it follows that136

E[ex] = exp(

E[x] +12var(x)

)

var(ex) = E[ex]2(exp(var(x))− 1)

Hence, we obtain

Et

[exp

(σr

κ

∫ T

t

((e−κ(T−s) − 1) + λ)dz(s)

)]

=exp

(12var

[∫ T

t

(σr

κ((e−κ(T−s) − 1) + λ)

)2

dz(s)

])

=exp

(12

∫ T

t

(σr

κ((e−κ(T−s) − 1) + λ)

)2

ds

)

=exp

(−

(3 + e−2(T−t)κ − 4e−(T−t)κ

)σ2

r

4κ3+

(1− e−(T−t)κ

)λσr

κ2

+(T − t)(σr − κλ)2

2κ2

)(3.23)

By substituting (3.23) into (3.22), we eventually obtain the zero-coupon bondprice formula

P (t, T ) = exp

(−1

2(T − t)λ2 +

(1− e−(T−t)κ

)σrλ

κ2

−(3 + e−2(T−t)κ − 4e−(T−t)κ

)σ2

r

4κ3+

(T − t)(σr − κλ)2

2κ2

−θ((T − t)κ + e−(T−t)κ − 1

)

κ−

(1− e−(T−t)κ

)

κr(t)

)(3.24)

This can be written in a shorter way by setting137

B(t, T ) =1κ

(1− exp(−κ(T − t))) (3.25)

and138

A(t, T ) =R(∞)(

1κ

(1− exp(−κ(T − t)))− (T − t))

− σ2r

4κ3(1− exp(−κ(T − t)))2 (3.26)

136 See Rinne (1997), p. 365.137 Svoboda (2004), p. 10.138 Svoboda (2004), p. 10.


where139

R(∞) =(

θ + λσr

κ− 1

2σ2

r

κ2

)(3.27)

Hence, the analytic solution for zero-coupon bond prices in the Vasicekterm structure model is

P (t, T ) = exp(A(t, T )−B(t, T )r(t)) (3.28)

3.4.3 Properties

First, we want to derive some properties of the short rate. It can be seen from(3.16), that the drift term is not constant but depends on the current levelof the short rate. θ is the long-term mean level of the short rate. Wheneverthe current short rate is above (below) the long-term mean level, the drift isnegative (positive). The parameter κ – the speed of adjustment – determineshow “fast” the short rate is pulled towards θ. Because of this behavior of theshort rate, all interest rates in the Vasicek model exhibit mean reversion.140

The future short rate r(T ) is (conditionally) normally distributed withmean141

Et[r(T )] = r(t) exp(−κ(T − t))︸︷︷︸w

+θ (1− exp(−κ(T − t)))︸︷︷︸1−w

(3.29)

and variance142

vart(r(T )) =σ2

r

2κ(1− exp(−2κ(T − t))) (3.30)

Expected future short rates are a weighted average of the current value ofthe short rate r(t) and the long-term mean value θ, the weights being w and1 − w respectively.143 For small values of κ the short rate reverts slowly tothe long-term mean and therefore more weight is given to the current valueof the short rate. For κ → 0, the weight w becomes 1 and hence the expectedvalue equals the current value.144 For T →∞ (and any value of κ) the meanapproaches θ.145

The effect of the parameters σr and κ on the variance are easily calculable.An increase (decrease) of σr increases (decreases) the variance of future short

139 Vasicek (1977), p. 186.140 Svoboda (2004), p. 9.141 Apply the expectations operator to Equation (3.20).142 This can be derived as well from Equation (3.20). See also Vasicek (1977), p. 185.143 See Tuckman (2002), p. 239.144 For large values of κ the short rate reverts fast to the long-term mean and therefore

less weight is given to the current value of the short rate. For κ →∞, the weightw becomes zero and hence the expected value equals the long-term mean level θ.

145 Munk (2004b), p. 157.

3.4 Vasicek (1977) 27

rates and an increase (decrease) in κ decreases (increases) it. This is quiteobvious, since an increase of κ makes the short rate revert faster to the long-term mean level θ and so the variance decreases.

A further inspection of (3.29) reveals the effect of the mean reversionfeature on short rate expectations. Depending on the relation between thecurrent short rate and the long-term mean level, the following cases can bedistinguished

• Et[r(T )] = r(t) if r(t) = θ. Whenever the current short rate is equal toits long term mean θ, the expectation is, that it doesn’t change,146 sinceover the long-term it must be pulled towards θ and this expected path isdisturbed only by uncorrelated zero-mean shocks.

• Et[r(T )] < r(t) if r(t) > θ. Whenever the current short rate is above θ,it is expected to decline.

• Et[r(T )] > r(t) if r(t) < θ. Whenever the current short rate is below θ,it is expected to rise.

The Vasicek model is Gaussian. So, like every Gaussian model, it assignspositive probabilities to negative values of the future short rate (and all otherfuture interest rates).147 Theoretically, this is a undesirable property of amodel, since negative interest rates are not possible in an arbitrage free mar-ket.148 The model is nevertheless free of arbitrage since holding cash is notallowed in the model.149 For “reasonable” parameter values, the possibility ofnegative future interest rates is fortunately quite small.150 In an influentialempirical study of the term structure Chan et al. (1992) found the followingparameters for the Vasicek model: κ = 0.1779, θ = 0.0865 and σr = 0.02.151

For these values, the probability of negative short rates is generally under5 %152

Table 3.3. Vasicek model: Probability of negative future short rates.

T 1 2 3 4 5

Pr(r(T ) < 0) 4.6 % 4.8 % 4.0 % 3.3 % 2.6 %

146 The drift term is equal to zero.147 See Martellini/Priaulet/Priaulet (2003), p. 390.148 People could hold cash and getting a zero return instead.149 Munk (2004b), p. 159.

150 It is N

(− Et[R(T )]√

vart(r(T ))

), where N(x) is the standard-normal cumulative probabil-

ity function, see Munk (2004b), p. 163.151 Chan et al. (1992), p. 1218.152 Calculations of the author. The probability of a negative future short rate in one

year is hence 4.6 % and that it is negative in five years is just 2.6 %.


Now, we have a closer look at the term structure of interest rates. WithP (t, T ) from (3.28) and R(t, T ) from (2.2) we can obtain the term structureof interest rates.153

R(t, T ) =− A(t, T )T − t

+B(t, T )T − t

r(t)

=R(∞) + (r(t)−R(∞))1

κ(T − t)(1− exp(−κ(T − t)))

+σ2

r

4κ3(T − t)(1− exp(−κ(T − t)))2 (3.31)

As can be seen, it is affine in r(t) and the Vasicek model hence belongsto the class of affine interest rate models. It can be shown, that the Vasicekmodel allows for normal, inverted and slightly humped term structures.154

The corresponding parameters are155

• For r(t) ≤ R(∞)− σ2r

4κ2 the term structure of interest rates is monotonicallyincreasing (i.e. normal)

• For r(t) ≥ R(∞) + σ2r

2κ2 it is monotonically decreasing (i.e. inverse)

• For R(∞)− σ2r

4κ2 ≤ r(t) ≤ R(∞) + σ2r

2κ2 it is humped

Next, we discuss how changes in the short rate affect the term structureof interest rates. This can be seen by taking the partial derivative of R(t, T )with respect to r(t)

∂R(t, T )∂r(t)

=B(t, T )T − t

=1− exp(−κ(T − t))

κ(T − t)(3.32)

Figure 3.1 shows the effect of changes in r on the term structure of interestrates (∂R

∂r ) as a function of maturity for different values of κ.

5 10 15 20 25 30T

0.2

0.4

0.6

0.8

1

¶R��¶r

Fig. 3.1. Vasicek model: ∂R∂r

as a function of maturity.

153 See Vasicek (1977), p. 186.154 See Vasicek (1977), p. 168.155 Vasicek (1977), pp. 186–187.

3.4 Vasicek (1977) 29

A rise in the short rate has no effect on the infinitely-long rate, becauseas can be seen from (3.27), it is independent of r. The effect of a change inthe short rate on the other interest rates is decreasing in T .156 The yield onan infinitely-lived bond is fixed at R(∞) and hence the short rate r(t) movesthe entire yield curve.157

Now, we are interested in the volatility structure of spot interest rates.In order to obtain it, we first have to derive the spot interest rate dynamics.With dr from (3.16) and R(t, T ) from (3.31), Ito’s lemma gives

dR(t, T ) = Rtdt + Rrdr +12Rrr(dr)2

= Rtdt +B(t, T )T − t

dr

=(

Rt +B(t, T )T − t

κ(θ − r))

dt +(

B(t, T )T − t

σr

)dz1

= µR(r, t, T )dt + σR(t, T )dz1

The expression σR(t, T ) describes the term structure of volatilities

σR(t, T ) =1− exp(−κ(T − t))

κ(T − t)σr (3.33)

Figure 3.2 contains a plot of the volatility structure of spot interest rates

5 10 15 20 25 30T

0.005

0.01

0.015

0.02

ΣRHt,TL

Fig. 3.2. Vasicek model: Volatility structure

It can be seen that the volatility declines when the term to maturity in-creases, i.e. short term rates are more volatile than long term rates. Empiricalstudies seem to confirm this volatility structure, but sometimes also humpedvolatility structures158 are observed in the marketplace.159 Humped volatility

156 Calculate the partial derivative ∂2R∂r∂T

. This expression is negative.157 See Holden (2005), p. 55.158 The volatility rises at first but then declines.159 See Golub/Tilman (2000), p. 89.


structures cannot result from a Vasicek model.160 The formula also impliesconstant interest rate volatilities.

A serious drawback of the model in a portfolio selection context is theperfect correlation of the spot rates

corr(R(t, T ), R(t, τ)) =cov(R(t, T ), R(t, τ))

std(R(t, T ))std(R(t, τ))

=cov

(B(t,T )T−t r(t), B(t,τ)

τ−t r(t))

std(

B(t,T )T−t r(t)

)std

(B(t,τ)τ−t r(t)

)

=B(t,T )T−t

B(t,τ)τ−t var(r(t))

B(t,T )T−t std(r(t))B(t,τ)

τ−t std(r(t))

= 1

Zero-coupon bond prices are nonlinear functions of the short rate.161

Therefore zero-coupon bonds of different maturities are also perfectly (be-cause there is only one source of randomness) but non-linearly correlated, i.e.the correlation coefficient is near but not equal to 1.162

A well-known extension to the Vasicek model is due to Hull/White(1990).163 One drawback of the original Vasicek model, is that it cannot per-fectly fit the initial term structure of interest rates.164 By making the drift ofthe short rate165 time-dependent, it is now possible to match any initial termstructure of interest rates by construction. In this thesis, we are interested inbond portfolio selection problems. The matching of an initial term structureof interest rates is in our application of no concern, since we assume that themodel term structure is equal to the observed initial term structure. Hence,we concentrate on the time-homogeneous models.166

3.5 Hull/White (1994)

3.5.1 Introduction

Hull/White (1994) propose a two-factor interest rate model based on Vasicek’sterm structure model (hereafter, HW2). The two factors are the short rate

160 The partial derivative ∂σR(τ)∂τ

is negative.161 See Equation (3.28).162 In the next section, we derive the HW2 model. It allows for non-perfect correla-

tions between interest rates.163 The so-called extended Vasicek model.164 Since the model has only a finite number of parameters, these cannot be chosen

in such a way as to match the prices of infinitely many bonds.165 Actually, the mean reversion level.166 The analysis can easily be extended to time-inhomogeneous models.

3.5 Hull/White (1994) 31

r and the mean-reversion level ε. This extension provides for a richer setof possible volatility structures and spot rate correlations. They assume thefollowing dynamics for the factors167

dr(t) = (θ + ε(t)− κrr(t))dt + σrdz1 (3.34)

dε(t) = −κεε(t)dt + σε%dz1 + σε

√1− %2dz2 (3.35)

where θ, σr, σε, κr > 0 and κε > 0 are constants.168 The Brownian motionsdz1 and dz2 are assumed to be uncorrelated and % is the correlation coefficientbetween the short rate r and the mean-reversion level ε.169

3.5.2 Derivation of Zero-Coupon Bond Prices

The HW2 model is also a special case of the general HJM framework. Themodel is a two-factor interest rate model, i.e. d = 2. The forward rate volatil-ities are assumed to be

s1(t, T ) =e(t−T )κr (κr − κε)σr −

(e(t−T )κr − e(t−T )κε

)%σε

κr − κε(3.36)

s2(t, T ) =

(−e(t−T )κr + e(t−T )κε)√

1− %2σε

κr − κε(3.37)

and the market prices of interest rate risk are constant

λ1(t) = λ1 (3.38)λ2(t) = λ2 (3.39)

The initial forward rate curve is given by

f(0, T ) = g(0, T ) + r(0)e−κrT + ε(0)e−κεT − e−κrT

κr − κε(3.40)

with g(0, T ) defined in Equation (A.6) in Appendix A.4.An application of the general zero-bond pricing equation (3.12) further

requires only an expression for the short rate f(T, T ). The short rate can beexplicitly calculated by application of Equation (3.15)

167 This formulation is equivalent to the original one with correlated Brownian mo-tions. See e.g. Brigo/Mercurio (2001), p. 134

168 In the original formulation θ is allowed to be a function of time. We restrictourselves to the special case θ(t) = θ.

169 To see this calculate the correlation between dε and dr

corr(dε, dr) =cov(dε, dr)

std(dε)std(dr)=

%σrσε

σrσε= %


f(T, T ) =f(0, T ) +∫ T

0

s1(u, T )

(∫ T

u

s1(u, s)ds− λ1

)du

+∫ T

0

s2(u, T )

(∫ T

u

s2(u, s)ds− λ2

)du

+∫ T

0

s1(u, T )dz1(u) +∫ T

0

s2(u, T )dz2(u) (3.41)

We insert (3.40) into (3.41). The resulting expression can be simplified furthersince it can be shown that

g(0, T ) +∫ T

0

s1(u, T )

(∫ T

u

s1(u, s)ds− λ1

)du

+∫ T

0

s2(u, T )

(∫ T

u

s2(u, s)ds− λ2

)du =

θ

κr(1− e−κrT )

Hence, we obtain the following formula for the short rate of interest in theHW2 model

f(T, T ) =θ

κr(1− e−κr(T−t)) + r(t)e−κr(T−t) + ε(t)

e−κε(T−t) − e−κr(T−t)

κr − κε

+∫ T

t

e(u−T )κr (κr − κε)σr −(e(u−T )κr − e(u−T )κε

)%σε

κr − κεdz1(u)

+∫ T

t

(−e(u−T )κr + e(u−T )κε) √

1− %2σε

κr − κεdz2(u) (3.42)

Now, we are in a position to derive the prices of zero-coupon bonds byapplying Equation (3.12). With (3.38) and (3.39) we obtain

P (t, T ) =Et

[exp

(−

∫ T

t

f(s, s)ds− 12λ2

1(T − t)− 12λ2

2(T − t)

+∫ T

t

λ1dz1(s) +∫ T

t

λ2dz2(s)

)](3.43)

First, we calculate the integral over the short rate

3.5 Hull/White (1994) 33

−∫ T

t

f(s, s)ds =

−∫ T

t

r(t)e−κr(s−t)ds−∫ T

t

θ

κr(1− e−κr(s−t))ds

−∫ T

t

ε(t)e−κε(s−t) − e−κr(s−t)

κr − κεds

−∫ T

t

∫ s

t

e(t−s)κr (κr − κε)σr −(e(t−s)κr − e(t−s)κε

)%σε

κr − κεdz1(u)ds

−∫ T

t

∫ s

t

(−e(t−s)κr + e(t−s)κε) √

1− %2σε

κr − κεdz2(u)ds (3.44)

The first three integrals in Equation (3.44) are Riemann integrals and can becalculated explicitly. By defining

B1(t, T ) =∫ T

t

e−κr(s−t)ds

=1− e−κr(T−t)

κr(3.45)

and

B2(t, T ) =∫ T

t

e−κε(s−t) − e−κr(s−t)

κr − κεds

=

(1− e(t−T )κε

)κr −

(1− e(t−T )κr

)κε

κr (κr − κε)κε(3.46)

we can write the result as

−∫ T

t

f(s, s)ds =

− r(t)B1(t, T )− ε(t)B2(t, T )− θ−1 + e−κr(T−t) + (T − t)κr

κ2r

−∫ T

t

∫ s

t

e(t−s)κr (κr − κε)σr −(e(t−s)κr − e(t−s)κε

)%σε

κr − κεdz1(u)ds

−∫ T

t

∫ s

t

(−e(t−s)κr + e(t−s)κε) √

1− %2σε

κr − κεdz2(u)ds (3.47)

The last two expressions in Equation (3.47) contain Riemann integrals andstochastic integrals. These expressions can be simplified to expressions con-taining stochastic integrals only by substitution and integration by parts.170

We apply the formula derived in Brigo/Mercurio (2001) and obtain the fol-lowing solution171

170 For a general integration by parts formula see Oksendal (1992), p. 46.171 Brigo/Mercurio (2001), p. 150.


−∫ T

t

r(s)ds =− r(t)B1(t, T )− ε(t)B2(t, T )

− θ−1 + e−κr(T−t) + (T − t)κr

κ2r

+∫ T

t

(−1+e(s−T )κε)%σε

κε+ (−1+e(s−T )κr )((κr−κε)σr−%σε)

κr

κr − κεdz1(s)

+∫ T

t

√1− %2

(−1+e(s−T )κε

κε− −1+e(s−T )κr

κr

)σε

κr − κεdz2(s)

We define

η(s) =

(−1+e(s−T )κε)%σε

κε+ (−1+e(s−T )κr )((κr−κε)σr−%σε)

κr

κr − κε+ λ1

and

ν(s) =

√1− %2

(−1+e(s−T )κε

κε− −1+e(s−T )κr

κr

)σε

κr − κε+ λ2

then the result can be written in a more readable form as

−∫ T

t

r(s)ds =− r(t)B1(t, T )− ε(t)B2(t, T )− θ−1 + e−κr(T−t) + (T − t)κr

κ2r

+∫ T

t

η(s)dz1(s) +∫ T

t

ν(s)dz2(s) (3.48)

We insert (3.48) into (3.43) and obtain the following expression for zero-coupon prices

P (t, T ) =

Et

[exp

(−r(t)B1(t, T )− ε(t)B2(t, T )− θ

−1 + e−κr(T−t) + (T − t)κr

κ2r

+∫ T

t

η(s)dz1(s) +∫ T

t

ν(s)dz2 − 12λ2

1(T − t)− 12λ2

2(T − t)

)]

The only random variables are the two stochastic integrals, hence

P (t, T ) = exp (−r(t)B1(t, T )− ε(t)B2(t, T )

−θ−1 + e−κr(T−t) + (T − t)κr

κ2r

− 12λ2

1(T − t)− 12λ2

2(T − t))

× Et

[exp

(∫ T

t

η(s)dz1(s) +∫ T

t

ν(s)dz2

)](3.49)

3.5 Hull/White (1994) 35

Next, we need to calculate the expectations of the exponential of the sum ofthe two stochastic integrals.172

Et

[exp

(∫ T

t

η(s)dz1(s) +∫ T

t

ν(s)dz2

)]

= exp

(12

∫ T

t

η(s)2ds +12

∫ T

t

ν(s)2ds

)(3.50)

We insert (3.50) into (3.49) and obtain

P (t, T ) = exp(−r(t)B1(t, T )− ε(t)B2(t, T )− θ

−1 + e−κr(T−t) + (T − t)κr

κ2r

−12λ2

1(T − t)− 12λ2

2(T − t) +12

∫ T

t

η(s)2ds +12

∫ T

t

ν(s)2ds

)

With

A(t, T ) =− θ−1 + e−κr(T−t) + (T − t)κr

κ2r

− 12λ2

1(T − t)− 12λ2

2(T − t)

+12

∫ T

t

η(s)2ds +12

∫ T

t

ν(s)2ds

and B1(t, T ) and B2(t, T ) from (3.45) and (3.46), this can be written in shortform as

P (t, T ) = exp (A(t, T )−B1(t, T )r(t)−B2(t, T )ε(t)) (3.51)

As can be seen from Equation (3.51), the HW2 model belongs to the affineclass as well.

3.5.3 Properties

First, we have a look at the short rate properties. Since this model builds onthe Vasicek model, it keeps many of its properties. The short rate is againnormally distributed with mean

Et[r(T )] =θ

κr(1− e−κr(T−t)) + r(t)e−κr(T−t)

+ ε(t)e−κε(T−t) − e−κr(T−t)

κr − κε(3.52)

172 The sum of the two stochastic integrals is normally distributed. If x is normally

distributed, then E[ex] = eE[x]+ 12 var(x), see Rinne (1997), p. 365. In this case,

E[x] = 0.


and variance173

vart(r(T )) = vart

(∫ T

t

s1(u, T )dz1(u) +∫ T

t

s2(u, T )dz2(u)

)

=∫ T

t

s1(u, T )2du +∫ T

t

s2(u, T )2du

=∫ T

t

(s1(u, T )2 + s2(u, T )2

)du (3.53)

With P (t, T ) from (3.51) and R(t, T ) from (2.2), we can obtain the termstructure of interest rates

R(t, T ) = −A(t, T )T − t

+B1(t, T )T − t

r(t) +B2(t, T )T − t

ε(t) (3.54)

With this expression for the term structure of interest rates, we are in aposition to analyze how changes in the factors influence the spot interestrates. As in the Vasicek model, we can obtain an expression for the interestrate with infinite maturity. The infinitely long rate is

R(∞) ≡ limT→∞

R(t, T )

=κε (θ + λ1σr) +

(%λ1 +

√1− %2λ2

)σε

κrκε− κ2

εσ2r + 2%κεσεσr + σ2

ε

2κ2rκ

2ε

It can be seen that it is a constant, i.e. this rate is neither influenced by r(t)nor ε(t). The effect of changes in the factors on the term structure can beobtained by calculating the respective partial derivatives ∂R(t,T )

∂r and ∂R(t,T )∂ε .

The first partial derivative

∂R(t, T )∂r

=B1(t, T )T − t

=1− exp(−κr(T − t))

(T − t)κr

is equivalent to the Vasicek model.174 Figure 3.3 shows the effect of changesin r(t) on the shape of the term structure of interest rates for different valuesof κr.

173 With s1(t, T ) and s2(t, T ) from (3.36) and (3.37).174 See Equation (3.32).

3.5 Hull/White (1994) 37

5 10 15 20 25 30T

0.2

0.4

0.6

0.8

1

¶R��¶r

Fig. 3.3. HW2 model: ∂R∂r


Hence changes in the short rate have a bigger influence on shorter ratesthan on longer rates. The infinitely long rate is not influenced by changes inthe short rate, since its level is independent of both random variables. Thesecond state variable, the mean reversion level, has a different impact on theterm structure of interest rates. The second partial derivative is

∂R(t, T )∂ε

=B2(t, T )T − t

=

(1− e(t−T )κε

)κr +

(−1 + e(t−T )κr)κε

(T − t)κr (κr − κε) κε

As can be seen from Figure 3.4, changes in ε(t) have no effect on the shortrate, the biggest effect on intermediate rates and smaller effect on long rates.

5 10 15 20 25 30T

0.1

0.2

0.3

0.4

0.5

0.6

¶R��¶Ε

Fig. 3.4. HW2 model: ∂R∂ε


Figures 3.3 and 3.4 resemble the factor analysis graphs of principal com-ponent analysis of the yield curve.175 The form of the factor influence is dif-ferent than empirical observations suggest, but the inclusion of a second (notperfectly correlated) factor clearly has the effect of introducing more realistic

175 See Litterman/Scheinkman (1991), p. 58.


term structure movements. When short rate changes and mean-reversion levelchanges are strongly correlated, one can expect unidirectional changes of thewhole term structure of interest rates since the partial derivatives have thesame sign. Principal component analysis of the term structure of interest rateson the other hand suggests that some factors have widely different impactson interest rates, i.e. changes in a factor move some spot rates up and somedown.

The spot rate volatility structure is also quite different. The Vasicek modelallows for a spot rate volatility structure that is decreasing in T . We alreadypointed out that some empirical studies suggest the existence of humpedvolatility structures.176 In contrast to the Vasicek model – as we will shownext – the HW2 model allows for these humped volatility structures.

With R(t, T ) from (3.54), Ito’s lemma can be used to determine the termstructure of volatilities177

dR(t, T ) =Rtdt + Rrdr + Rεdε +12Rrr(dr)2 +

12Rεε(dε)2 + Rrε(dr)(dε)

=Rtdt +B1(t, T )T − t

dr +B2(t, T )T − t

dr

=(

Rt +B1(t, T )T − t

(θ + ε(t)− κrr(t))− B2(t, T )T − t

κεε(t))

dt

+(

B1(t, T )T − t

σr +B2(t, T )T − t

σε%

)dz1 +

B2(t, T )T − t

σε

√1− %2dz2

=µR(r, ε, t, T )dt + σ1,R(t, T )dz1 + σ2,R(t, T )dz2

With

σ1,R(t, T ) =B1(t, T )T − t

σr +B2(t, T )T − t

σε% (3.55)

σ2,R(t, T ) =B2(t, T )T − t

σε

√1− %2 (3.56)

it follows that the volatility of spot interest rates is

σR(t, T ) =√

σ1,R(t, T )2 + σ2,R(t, T )2 (3.57)

Figure 3.5 provide a graphical representation of the spot rate volatility struc-ture.178

176 See Golub/Tilman (2000), p. 89.177 dr and dε from (3.34) and (3.35).178 The following parameter values have been chosen: % = 0.5, κr = 0.5, κε =

0.1, σr = 0.01, σε = 0.04.

3.6 Summary and Conclusion 39

5 10 15 20 25 30T

0.015

0.02

0.025

0.03

0.035

0.04

ΣRHt,TL

Fig. 3.5. HW2 model: Volatility structure.

Lastly, the introduction of a second factor has an important effect on thecorrelation structure of spot rates (or zero-coupon bond prices).

corr(R(t, T ), R(t, τ))

=cov(R(t, T ), R(t, τ))

std(R(t, T ))std(R(t, τ))

=cov

(B1(t,T )

T−t r(t) + B2(t,T )T−t ε(t), B1(t,τ)

τ−t r(t) + B2(t,τ)τ−t ε(t)

)

std(

B1(t,T )T−t r(t) + B2(t,T )

T−t ε(t))

std(

B1(t,τ)τ−t r(t) + B2(t,τ)

τ−t ε(t))

This expression is not necessarily equal to 1. Therefore, spot rates (and zero-coupon bond prices) of different maturities are not perfectly correlated in theHW2 model. This is especially interesting in a portfolio context, as we willsee in the next chapter.

The original formulation of the model assumed that θ is not constant buta deterministic a function of time.179 With this extension, the model is ableto match any particular initial term structure. Furthermore, the volatilityfunctions σr and σε can be made deterministic functions of time as well andhence the model could match any volatility structure. But this is generallyproblematic since doing so results in unrealistic deterministic movements ofthe volatility structure over time.180

3.6 Summary and Conclusion

In this chapter we derived the general HJM framework using the stochasticdiscount factor approach. Furthermore, we derived and analyzed two specialcases, the one-factor Vasicek (1977) model and the two-factor HW2 model.

179 See Hull/White (1996), p. 334.180 See Hull/White (1996), p. 369.


From a portfolio selection perspective, the HW2 model has the advantageof capturing the real-world interest rate correlations better. In the Vasicekmodel, all interest rates are perfectly correlated. Also the possible term struc-ture movements in the HW2 model are more realistic than in the Vasicekmodel.

In Chapters 4 and 5 we incorporate the Vasicek (1977) and the HW2models into bond portfolio optimization problems.

4

Static Bond Portfolio Optimization

4.1 Introduction

This chapter describes static bond portfolio optimization based on the mean-variance framework of Markowitz (1952). This theoretical approach for bondportfolio selection was first introduced by Wilhelm (1992). We show why andhow the Markowitz model has to be adapted in order to be useful for theselection of bond portfolios. After deriving the adjusted portfolio selectionproblem, we apply it to the Vasicek and the HW2 model, we presented inChapter 3.181 Furthermore, numerical examples highlight potential problemsand inner workings of the model. In the last part of the chapter, this theo-retical bond portfolio selection model is compared – by means of numericalexamples – to well-known active and passive portfolio selection methods usedin practice.182

4.2 Static Bond Portfolio Selection in Theory

4.2.1 A Short Review of Modern Portfolio Theory

Modern portfolio theory183 – introduced by Markowitz in 1952184 – is the“cornerstone of modern asset management”.185 It is a static model in thesense that the investor is assumed to construct a portfolio today and to sell itat a later date (the investment horizon). Between today and the investmenthorizon, there is no portfolio re-balancing.186 Realistically portfolio choice

181 Wilhelm (1992) applies this framework to the term structure model by Cox/Ingersoll/Ross (1985).

182 A definition of active and passive strategies follows later.183 A well-known textbook covering most aspects of modern portfolio theory is El-

ton et al. (2003).184 Markowitz (1952).185 Scherer (2002), p. 1.

42 4 Static Bond Portfolio Optimization

should be analyzed in a multi-period setting. But the problem then becomesfar more complicated and less tractable. Theoretically, a multi-period portfolioselection problem reduces to a sequence of single-period problems (so-calledmyopic portfolio choice) only if187

• returns are independent and identically-distributed random variables• the investor has a utility function with constant relative risk aversion, i.e.

RRA is independent of wealth.

A second assumption of the model is that the investor cares only aboutexpected terminal portfolio wealth (or portfolio return) and variance of termi-nal portfolio wealth (or variance of portfolio return). Economic theory on theother hand claims, that a rational investor maximizes his expected utility ofterminal wealth (or consumption).188 It has been shown, that the investor’sexpected utility of terminal wealth is a function of the mean and the vari-ance of terminal wealth only if the preferences of the investor are governedby a quadratic utility function or asset prices are multi-variate normally dis-tributed.189

With these assumptions, the problem of the investor is to minimize theportfolio variance given an expected terminal wealth and a budget constraint.

minN

var(WT ) (4.1)

s.t. E[WT ] = WT

n∑

i=1

NiPi = W0

where W0 is the initial and WT is terminal wealth, Ni is the quantity ofasset i in the portfolio, Pi is the price of asset i, E is the expectation andvar is the variance operator. The desired expected terminal wealth of theinvestor is denoted by WT . Other constraints could be imposed as well, e.g.short-sale constraints.190 But closed-form solutions are only available for theunrestricted case.191 For all other cases, there exists a fast algorithm – thecritical line method – for computing the efficient frontier in a parameterizedform.192 For a more recent exposition of the critical line method, that includes

186 In the real world, the investor will most likely rebalance his portfolio betweenthese two dates. But this is not reflected in the model.

187 See Mossin (1968), p. 228.188 See for example Theorem 3 in Ingersoll (1987), p. 31.189 See Ingersoll (1987), p. 96.190 Ni ≥ 0 ∀ i = 1, . . . , n191 The general solution to the mean-variance portfolio selection problem is well

known and will not be given here since – as will be shown in the next section –the problem has first to be adapted to portfolios of bonds.

192 The critical line method was developed by Markowitz and is described inMarkowitz (1956) and Markowitz (1959).

4.2 Static Bond Portfolio Selection in Theory 43

modern results from operations research, see Rudolf (1994), Markowitz/Todd(2000) and Mertens (2006).

4.2.2 Application to Bond Portfolios

The formulation of the portfolio selection problem in (4.1) applies in principleto all tradeable assets. But an application to the selection of bonds createsdifficulties because the differences between stocks and bonds have to be takeninto account since they affect the calculation of the terminal wealth WT .

One of the biggest differences between stocks and bonds is the finite matu-rity of the latter asset class. This difference introduces an important problemto the above formulation of the optimization program. Bonds with a maturityless than the investment horizon won’t exist at the investment horizon any-more. Hence, we need an assumption about the reinvestment opportunities forcash flows – face value or coupons – received before the investment horizonT .193 This reinvestment assumption must meet a certain requirement: it hasto be possible with the information available at the time of the cash flows todecide on an optimal policy. Hence we can’t assume that we rebalance theportfolio every time a cash flows occurs, since this is a classical dynamic pro-gramming problem and in such a problem the decision to be taken at time thas to anticipate later decisions.194 For the static mean variance framework,we need a reinvestment assumption that doesn’t need to anticipate futureportfolio rebalancing decisions. Wilhelm (1992) assumed – in accordance withthe standard duration strategy – that all cash flows received at time t < Tshall be invested at the current spot interest rate R(t, T ) until the investmenthorizon T .195, 196 In order to execute this strategy optimally at time t no an-ticipation of future decisions at times ti with t < ti < T is necessary. Otherreinvestment assumptions are possible but this one seems to be a reasonableapproximation to reality.197

In our analysis, we want to restrict our attention to portfolios of zero-coupon bonds. This is no limitation since we can think of coupon bonds simplyas portfolios of zero-coupon bonds.198 In this regard zero-coupon bonds can

193 See Wilhelm (1992), p. 216.194 See Wilhelm (1992), p. 216.195 See Wilhelm (1992), pp. 216–217.196 We hence assume that suitable zero-coupon bonds exist on all cash-flow dates.197 In Chapter 5 we will consider a dynamic portfolio selection model where the

portfolio can be rebalanced continuously.198 Example: A 2 year 5% coupon bond with a face value of 100 can be regarded

as a portfolio of a one year zero-coupon bond with face value 5 and a two yearzero-coupon bond with face value 105.


be thought of as the basic portfolio building blocks.199 This approach can ofcourse be easily generalized to coupon bonds.200

Let the investment universe201 consist of zero-coupon bonds of differentmaturities. The longest maturity of all zero-coupon bonds is denoted by τ .202

There exists one bond for each maturity date 1, 2, ..τ − 1, τ .At time t = 0 the investor allocates his initial wealth W0 to the τ zero-

coupon bonds.

W0 =τ∑

t=1

NtP (0, t)

where Nt denotes the purchased quantity of the zero-coupon bond with matu-rity date t and current price P (0, t). The τ zero-coupon bonds can be dividedinto one riskless and τ−1 risky assets. An investment in the zero-coupon bondwith maturity T is riskless. All other zero-coupon bonds are risky investmentsand are – for notational convenience – combined in the holdings vector N andthe price vector P0

W0 = N ′P0 + NT P (0, T ) (4.2)

with203

N ′ = (N1, . . . , NT−1, NT+1, . . . , Nτ )

P0′= (P (0, 1), . . . , P (0, T − 1), P (0, T + 1), . . . , P (0, τ))

NT denotes the number of T -maturity bonds the investor purchases at timezero. Next, we derive the terminal wealth of the investor. An investment ofNT P (0, T ) in the riskless T -maturity bond at time zero grows to NT at time T .Holdings of zero-coupon bonds with a maturity greater than the investmenthorizon are at time T simply worth the sum of the (arbitrage-free) pricesof the individual zero-coupon bonds. Holdings of zero-coupon bonds with amaturity less than T are more difficult to value at T . The face value of thesezero-coupon bonds is reinvested at time t < T at the current spot rate R(t, T )until the investment horizon. Hence, the terminal wealth is204

WT =T−1∑t=1

Nt exp((T − t)R(t, T )) + NT +τ∑

t=T+1

Nt exp(−(t− T )R(T, t))

With R(t, T ) from (2.2) it follows that199 Furthermore, this approach is more akin to the real world approach. In prac-

tice one rarely observes “government bond picking” but the portfolio managersposition themselves appropriately on the yield curve.

200 For a formulation with coupon bonds, the interested reader is referred to Wilhelm(1992).

201 Or investment opportunity set.202 Generally τ will be greater than T but this is not required.203 The symbol ′ denotes transpose.204 See Wilhelm (1992), p. 217.


WT =T−1∑t=1

Nt1

P (t, T )+ NT +

τ∑

t=T+1

NtP (T, t)

Let

PT =(

1P (1, T )

, . . . ,1

P (T − 1, T ), P (T, T + 1), . . . , P (T, τ)

)′

then the terminal wealth can be written in vector notation as follows

WT = N ′PT + NT (4.3)

In the mean-variance framework, the investor cares only about the expectedvalue and variance of terminal wealth. We apply the expectation operator toEquation (4.3) and obtain205

E[WT ] =T−1∑t=1

NtE

[1

P (t, T )

]+ NT +

τ∑

t=T+1

NtE [P (T, t)]

= N ′E[PT

]+ NT (4.4)

and the variance of terminal wealth is

var(WT ) =T−1∑t=1

T−1∑s=1

NtNscov(

1P (t, T )

,1

P (s, T )

)

+τ∑

t=T+1

τ∑

s=T+1

NtNscov (P (T, t), P (T, s))

+ 2T−1∑t=1

τ∑

s=T+1

NtNscov(

1P (t, T )

, P (T, s))

=N ′CN (4.5)

where C is the covariance matrix. Entries of C are denoted by σi,j withi, j = 1, . . . , τ − 1. The matrix C is defined as follows206

σi,j =

cov(

1P (i,T ) ,

1P (j,T )

)for i = 1, 2, . . . , T − 1

j = 1, 2, . . . , T − 1cov

(1

P (i,T ) , P (T, j + 1))

for i = 1, 2, . . . , T − 1j = T, T + 1, . . . , τ − 1

cov(P (T, i + 1), 1

P (j,T )

)for i = T, T + 1, . . . , τ − 1

j = 1, 2, . . . , T − 1cov (P (T, i + 1), P (T, j + 1)) for i = T, T + 1, . . . , τ − 1

j = T, T + 1, . . . , τ − 1

(4.6)

205 See Wilhelm (1992), p. 217.206 We assume that τ > T .


For a maximum maturity date of τ = 5 and an investment horizon of T = 3the covariance matrix C would be given by

var(

1P (1,3)

)cov

(1

P (1,3) ,1

P (2,3)

)cov

(1

P (1,3) , P (3, 4))

cov(

1P (1,3) , P (3, 5)

)

· · · var(

1P (2,3)

)cov

(1

P (2,3) , P (3, 4))

cov(

1P (2,3) , P (3, 5)

)

· · · · · · var (P (3, 4)) cov (P (3, 4), P (3, 5))· · · · · · · · · var (P (3, 5))

With the initial wealth from (4.2), the expected terminal wealth from (4.4)and the variance of terminal wealth from (4.5) the bond portfolio selectionproblem can be stated as follows207

minN

12N ′CN (4.7)

s.t. N ′E[PT

]+ NT = WT (4.8)

N ′P0 + NT P (0, T ) = W0 (4.9)

We can combine Equations (4.8) and (4.9) and obtain

minN

12N ′CN (4.10)

s.t.W0

P (0, T )︸︷︷︸(∗)

+N ′(

E[PT

]− 1

P (0, T )P0

)

︸︷︷︸(∗∗)

= WT

where (∗) denotes the riskfree compounded initial wealth and (∗∗) denotes thevector of risk premia. This problem looks very much identical to the equityformulation208, but the bond specific adjustments are “hidden” in the vectorsP0 and E

[PT

]as well as in the covariance matrix C. In order to calculate

mean-variance-efficient bond portfolios, we therefore need the following inputdata209

207 As usual we minimize half the variance.208 This was intended.209 See Wilhelm (1992), p. 217.


Table 4.1. Bond portfolio optimization: Input parameters.

Expression Parameter Description

E[P (T, t)] t = T + 1, . . . , τ expected discount factors attime T for all maturities greaterthan T

E[

1P (t,T )

]t = 1, 2, . . . , T − 1 expected accrual factors from t

to T

cov(

1P (t,T )

, 1P (s,T )

)s, t = 1, 2, . . . , T − 1 covariances between different

accrual factors

cov (P (T, t), P (T, s)) s, t = T + 1, . . . , τ covariances between discountfactors at T

cov(

1P (t,T )

, P (T, s))

t = 1, 2, . . . , T − 1 ands = T + 1, . . . , τ

covariances between accrual fac-tors and discount factors

Problem (4.10) is a quadratic optimization problem with one equality con-straints. It can be solved by differentiating the Lagrange function with re-spect to N .210

d

dN

(12N ′CN + λ

(WT − N ′

(E

[PT

]− 1

P (0, T )P0

)− W0

P (0, T )

))= 0

where 0 is a vector of zeros. We obtain the following solution for the zero-coupon bond holdings vector

N = λ

(C−1E

[PT

]− 1

P (0, T )C−1P0

)

︸︷︷︸Tobin fund y

= λy (4.11)

The efficient portfolio of risky assets is a multiple of the so-called Tobin fund y.Next, we insert the optimal solution N in the equality constraint and solve forλ. This yields the optimal value for the individual parameter λ as a functionof initial and expected future wealth.

λ =WT − W0

P (0,T )(E

[PT

]− 1

P (0,T ) P0

)′C−1

(E

[PT

]− 1

P (0,T ) P0

) (4.12)

With the solution for λ and N , the bond portfolio selection problem issolved. If WT = W0

P (0,T ) , i.e. the desired expected wealth in T is equal to theriskless compounded wealth, then from Equation (4.12) it can be seen thatλ = 0 and the entire wealth is invested in the T -maturity zero-coupon bond.The entire initial wealth is invested in the risky assets if λ = W0

P0.y, the resulting

portfolio is called the tangency portfolio.210 See Bronstein et al. (1999), p. 396.


Ntan =W0

P0.yy (4.13)

The problem of obtaining the necessary parameters is discussed in the nextsection.

4.2.3 Obtaining the Parameters

In equity portfolio selection, the necessary input parameters (expected returnsand covariances of returns) can be obtained by analyzing the time series ofthe assets. One approach is to impose no structure on the expected returnsand covariances of returns and use sample mean and sample covariances asinputs.211 At the other end of the spectrum are single- or multi-index modelsfor the asset returns.212 A compromise between these two approaches areso-called (James/Stein) shrinkage estimation techniques.213

In bond portfolio optimization, the problem of obtaining the necessaryinput parameters is more difficult as will become clearer from the followingcomments. We examine every parameter group from Table 4.1 concerningpossible estimation procedures.

The expected zero-coupon bond prices at the investment horizon E [P (T, t)]must be known. Using simple time series analysis is not recommended, sincebonds have finite maturities and promise to pay the face value at maturity,so the probability distribution depends on time to maturity.214 Consider a5-year zero-coupon bond at issuance. The price at the end of the year is ran-dom. Consider the same bond 4 years later. Then it has become a 1-yearzero-coupon bond and so the price at the end of the year is non-random, be-cause it pays the face value at maturity. The return distribution is hence timedependent. A T -year zero-coupon bond today becomes a (T − 1)-year zero-coupon bond in one year’s time, hence using the whole time series for expectedreturn or variance estimation is not advised, since these price observations arenot for the same asset.215 The problem can be mitigated by analyzing artifi-cial time series for so-called constant maturity bonds.216 Since knowledge ofthe expected zero-bond prices (discount curve) at the investment horizon isequivalent to the knowledge of the expected term structure of interest rates,one could also derive these input parameters by estimating the term structure

211 This is proposed in the classic book by Markowitz (1959).212 Because they impose a fixed structure on the asset returns. This approach is

explored in Chapter 7 and 8 in Elton et al. (2003).213 This approach was suggested by James/Stein (1961). Its application to portfolio

selection has been explored in several papers, e.g. Jorion (1986) and Ledoit/Wolfe(2004).

214 See Wilhelm (1992), p. 213.215 At the beginning for a T -year zero-coupon bond and at the end for 1-year zero-

coupon bond.216 See Elton et al. (2003), p. 545.


of interest rates for each day in a certain historical period and estimating thesample distribution parameters. Any of the classical term structure theoriespresented in Chapter 2.5 could as well be used to obtain these parameters.217

Furthermore, dynamic term structure models can be employed. The expectedaccrual factors E

[1

P (t,T )

]for every date before the investment horizon, could

be obtained in the same way.The covariances cov (P (T, t), P (T, s)) can be obtained by analyzing con-

stant maturity bonds as described above or using dynamic term structuretheories. Classical term structure theories yield no usable information.

The problem becomes more difficult when we consider cov(

1P (t,T ) ,

1P (s,T )

)

and cov(

1P (t,T ) , P (T, s)

). Here the covariance between functions of spot inter-

est rates (i.e. accrual and discount factors) for different dates or with differentmaturity must be derived. Term structure estimation gives just a snapshot ofthe current term structure and no indication of how the term structure movesover time. So it is impossible to get information about the co-movements be-tween term structures at different dates. Only dynamic term structure modelsdeliver these kind of information.

We can conclude that the input parameters for the bond portfolio opti-mization problem in (4.10) can only be obtained by employing dynamic termstructure models and hence imposing a fixed structure on the bond market. Ifthe investment horizon is the next possible date, then there exist other possi-bilities for obtaining the parameters and dynamic term structure models arenot the only choice.218

In theory, every dynamic term structure model could be used for obtain-ing the bond portfolio selection parameters. In practice, there is a trade-offbetween realism and analytical tractability.

In Chapter 3 we introduced two special cases of the general HJM frame-work, the Vasicek model and the HW2 model. These two models are membersof the Gaussian affine class – first studied by Langetieg (1980).219 In ouranalysis we want to restrict our attention to these two models.220

In a Gaussian affine model, zero-coupon bond prices are of the form221

217 See Elton et al. (2003), p. 545.218 Elton et al. (2003) suggests classical term structure theories and single- or multi-

index models, see Elton et al. (2003), pp. 540–546.219 The affine models were proposed by Duffie/Kan (1996).220 The advantage of the above models is their analytical tractability. If no analyt-

ical solution for zero-coupon bond prices exists, then the parameters cannot becalculated explicitly and we might have to resort to approximations to the nor-mal distribution. Wilhelm (1992) assumed that the term structure follows theCox/Ingersoll/Ross (1985) model. In this model the short rate is chi-squared dis-tributed. He then used an approximation to the normal distribution in order tocalculate the bond portfolio selection parameters.

221 Cairns (2004), p. 103.


P (t, T ) = exp(A(t, T )−k∑

j=1

Bj(t, T )xj(t))

= exp(A(t, T )−B(t, T )′x(t)) (4.14)

where x(t) is a k-dimensional vector of state variables. Furthermore it is as-sumed that the state variables are k-dimensionally normally distributed withmean E[x(t)] and covariance matrix COV(x(t), x(t)).222 Since the short rateis a linear combination of these state variables, the short rate is normallydistributed as well.

With the equation for zero-coupon bond prices in (4.14) and the assump-tion of k-dimensionally normally distributed state variables, we obtain thefollowing results for the bond portfolio selection parameters.223 The expecteddiscount factors (zero-coupon bond prices at time t with maturity T ) are

E [P (t, T )] =E [exp(A(t, T )−B(t, T )′x(t))]= exp (A(t, T )−B(t, T )′E[x(t)]

+12B(t, T )′COV(x(t), x(t)′)B(t, T )

)(4.15)

The expected accrual factors from t to T can be given by

E

[1

P (t, T )

]=E [exp(−A(t, T ) + B(t, T )′x(t))]

= exp (−A(t, T ) + B(t, T )′E[x(t)]

+12B(t, T )′COV(x(t), x(t)′)B(t, T )

)(4.16)

The covariances between discount factors at time t for maturities T and τ are

cov (P (t, T ), P (t, τ)) =E[P (t, T )]E[P (t, τ)]

×(eB(t,T )′COV(x(t),x(t)′)B(t,τ) − 1

)(4.17)

The covariances between accrual factors for maturity T at times t and τ canbe calculated as follows

cov(

1P (t, T )

,1

P (τ, T )

)=E

[1

P (t, T )

]E

[1

P (τ, T )

]

×(eB(t,T )′COV(x(t),x(τ)′)B(τ,T ) − 1

)(4.18)

The covariances between accrual factors at time t with maturity T and dis-count factors at time T with maturity τ are given next

222 See Langetieg (1980), p. 82.223 See Table 4.1. If x is normally distributed, then E[ex] = eE[x]+ 1

2 var(x) andvar(ex) = E[ex]2(evar(x) − 1), see Rinne (1997), p. 365


cov(

1P (t, T )

, P (T, τ))

=E

[1

P (t, T )

]E [P (T, τ)]

×(e−B(t,T )′COV(x(t),x(T )′)B(T,τ) − 1

)(4.19)

In order to calculate the above expressions, we only need the term struc-ture model dependent functions A(t, T ) and B(t, T ) and the distributionalparameters of the state variables. For the Vasicek and the HW2 model, theseexpression have already been derived in Chapter 3.

In Section 4.2.4 and 4.2.5 we analyze static bond portfolio optimizationinside the Vasicek model and the HW2 model by means of numerical examples.

4.2.4 One-Factor Vasicek (1977) Model

We assume the existence of a bond market where zero-coupon bonds of ma-turities 1, . . . , τ = 10 trade. This provides for a large but still manageableset of bonds. The investment horizon of the investor is denoted by T with1 ≤ T ≤ τ . The investor can therefore choose between nine risky zero-couponbonds and one riskless zero-coupon bond (the T -maturity bond).

We consider two investors who differ in their investment horizon: short-term investment horizon (T = 1) and long-term investment horizon (T = 5).The difference is of importance, since in the short-term case the portfoliovalue at the investment horizon is solely determined by the term structure atthe investment horizon. In the long-term case, on the other hand, it is alsoinfluenced by the term structures before the investment horizon.

We introduced the Vasicek (1977) model already in Chapter 3.4. The shortrate is the only state variable and it is normally distributed. Hence, the generalportfolio optimization input parameters from (4.15) to (4.19) can be specifiedas follows

E

[1

P (t, T )

]= exp

(−A(t, T ) + B(t, T )E[r(t)] +

12B(t, T )2var(r(t))

)

E [P (t, T )] = exp(

A(t, T )−B(t, T )E[r(t)] +12B(t, T )2var(r(t))

)

cov(

1P (t, T )

,1

P (τ, T )

)=

E

[1

P (t, T )

]E

[1

P (τ, T )

](eB(t,T )B(τ,T )cov(r(t),r(τ)) − 1

)

cov (P (t, T ), P (t, τ)) = E[P (t, T )]E[P (t, τ)](eB(t,T )B(t,τ)var(r(t)) − 1

)


cov(

1P (t, T )

, P (T, τ))

=

E

[1

P (t, T )

]E [P (T, τ)]

(e−B(t,T )B(T,τ)cov(r(t),r(T )) − 1

)

with A(t, T ) and B(t, T ) defined in Equations (3.26) and (3.25). In order tocalculate the above expressions, we additionally need the expected value andthe variance of the short rate and the covariances between short rates atdifferent times. These have already been calculated in (3.29) and (3.30)

Et[r(T )] = r(t) exp(−κ(T − t)) + θ(1− exp(−κ(T − t)))

vart(r(T )) = σ2r

(1− exp(−2κ(T − t))

2κ

)

covt(r(T ), r(τ)) = σ2r

(1− exp(−2κ(min(T, τ)− t))

2κ

)

The term structure parameters – the current level of the short rate r(0), themean reversion speed κ, the short rate volatility σr, the mean reversion levelθ and the market price of interest rate risk λ – are normally chosen in sucha way to match as closely as possible the current term structure of interestrates (perhaps subject to some economically sensible constraints for certainparameters).224 For our numerical example, we assume the following valuesfor the parameters225

Table 4.2. Vasicek model: Parameter values for numerical example.

Parameter Value

r(0) 0.0258θ 0.024κ 0.1668σr 0.0153λ 0.2126

As can be seen from Figure 4.1 these parameters describe a normal termstructure, i.e. monotonically increasing spot interest rates.

Short-term investment horizon

The short rate is the only systematic risk factor in the Vasicek model.226

Therefore, we first give the distributional properties of the risk factor and224 See Munk (2004b), p. 181.225 These values give a close approximation of the German government term structure

in January 2006.226 There are no idiosyncratic risk factors.


then deduce the distributional parameters of the assets (the zero-coupon bondprices). The current value of the short rate determines the current shape ofthe term structure of interest rates227, so the shape of the term structure atthe investment horizon is also determined by the future value of the shortrate only. The short rate at time T = 1 is normally distributed with mean228

E[r(1)] = 0.0255235 and standard deviation229 std(r(1)) = 0.0141084. Giventhe distribution of the short rate we can determine the distribution of allinterest rates at time T .230 Spot interest rates are normally distributed withmean231

E0[R(t, T )] = −A(t, T )T − t

+B(t, T )T − t

E0[r(t)]

and variance

var0(R(t, T )) =(

B(t, T )T − t

)2

var0(r(t))

From Figure 4.2 we observe that short term rates are more volatile thanlonger rates.232

2 4 6 8 10T

0.026

0.028

0.03

0.032

0.034

RH0,TL

Fig. 4.1. Vasicek: Term structure

2 4 6 8T

0.002

0.004

0.006

0.008

0.01

0.012

0.014

ΣRH0,TL

Fig. 4.2. Vasicek: Volatility structure.

An important feature of any term structure model is the ability to simul-taneously match the current term structure of interest rates and the covari-ance structure of interest rates. Matching the current term structure wouldbe possible by introducing a time-dependent drift as in the extended Vasicekmodel.233 But matching the covariance structure is impossible. Spot interestrates of different maturities are functions of the short rate only. In the Vasicek

227 Given the constant parameters.228 See Equation (3.29).229 See Equation (3.30).230 See Equation (3.31) for an expression of R(t, T ) in terms of r(t).231 Apply the expectations and variance operator to Equation (3.31).232 This is empirically observable in the fixed income markets, but humped shapes

– which cannot result from a Vasicek model – are also common, see Golub/Tilman (2000), p. 89.

233 Hull/White (1990).


model, this function is affine, so spot interest rates are perfectly (linearly) cor-related. Empirical observation suggests that spot rates of different maturitiesare positively but not perfectly correlated.234 The perfect correlation betweenspot interest rates is one of the most serious drawbacks of one-factor (affine)interest rate models.235

Zero-coupon bond prices are non-linear functions of spot rates.236 Theircorrelation structure is therefore quite similar to the spot rate correlationstructure. In the Vasicek model, they are perfectly – but non-linearly – cor-related. The correlation matrix Ψ of the risky zero-coupon bonds is in ournumerical example237

Ψ =

1 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.990.99 1 0.99 0.99 0.99 0.99 0.99 0.99 0.990.99 0.99 1 0.99 0.99 0.99 0.99 0.99 0.990.99 0.99 0.99 1 0.99 0.99 0.99 0.99 0.990.99 0.99 0.99 0.99 1 0.99 0.99 0.99 0.990.99 0.99 0.99 0.99 0.99 1 0.99 0.99 0.990.99 0.99 0.99 0.99 0.99 0.99 1 0.99 0.990.99 0.99 0.99 0.99 0.99 0.99 0.99 1 0.990.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1

(4.20)

Future zero-coupon bond prices are lognormally distributed because theshort rate is normally distributed.238 The expected values and standard de-viations of zero-coupon bond prices of different maturities (1, . . . , 10) at timeT = 1 are given in the next table.

Table 4.3. Vasicek model: Distribution of zero-coupon bond prices.

Ti 1 2 3 4 5 6 7 8 9 10

E0[P (1, Ti)] 1.000 0.974 0.946 0.917 0.888 0.858 0.829 0.801 0.772 0.745

std0(P (1, Ti)) 0.000 0.013 0.023 0.031 0.037 0.041 0.044 0.047 0.048 0.049

With current prices P (0, Ti) we can calculate the (continuously com-pounded) expected returns of the zero-coupon bonds over the next period.

234 See Golub/Tilman (2000), p. 89.235 See Martellini/Priaulet/Priaulet (2003), p. 391.236 See Equation (3.28).237 The correlation matrix can be determined from the covariance matrix that was

defined in Equation (4.6). We truncated the correlations at two decimal placesfor presentational purposes.

238 See Equation (3.28).


Table 4.4. Vasicek model: Expected holding period returns.

Ti 1 2 3 4 5 6 7 8 9 10

Exp. ret. (%) 2.716 2.975 3.18 3.345 3.477 3.584 3.671 3.743 3.802 3.85

In order to gain some intuition about possible portfolio returns, we performa simple scenario analysis. The different scenarios are determined by the valueof the short rate at time 1 only.239 When the short rate rises (falls), longer-term bonds perform worse (better) than short-term bonds. When the termstructure stays the same (i.e. r(1)=r(0)=0.025) then every zero-coupon bondwith the maturity Ti earns the forward rate between Ti − 1 and Ti.240

Table 4.5. Vasicek model: Zero-coupon bond returns in different scenarios.

Ti 1 2 3 4 5 6 7 8 9 10

r(1) = 0.030 2.716 2.562 2.419 2.288 2.17 2.066 1.974 1.894 1.824 1.764

r(1) = 0.025 2.716 3.023 3.269 3.468 3.63 3.762 3.87 3.959 4.033 4.094

r(1) = 0.020 2.716 3.483 4.119 4.648 5.089 5.457 5.766 6.024 6.241 6.423

r(1) = 0.015 2.716 3.944 4.97 5.829 6.549 7.153 7.661 8.089 8.449 8.753

r(1) = 0.010 2.716 4.404 5.82 7.009 8.008 8.849 9.557 10.15 10.66 11.08

r(1) = 0.005 2.716 4.865 6.67 8.189 9.467 10.54 11.45 12.22 12.87 13.41

These tables were provided for assessing whether or not the term structureparameter values in Table 4.2 are sensible. Huge percentage gains of individualzero-coupon bonds would have attracted attention in Table 4.5. We concludethat the parameters provide for realistic zero-coupon bond returns.

Next, we turn our attention to mean-variance efficient portfolios and obtainthe tangency portfolio. For an initial wealth of W0 = 1 unit of account, thetangency portfolio Ntan is241

Ntan =

19.06−155.91735.31−2198.364312.56−5543.494497.95−2088.91422.85

(4.21)

239 Of course the probability for the short rate to be exactly x is equal to zero, butnevertheless we give here 5 scenarios for 5 different values of the short rate.

240 See e.g. Ilmanen (1995), p. 4.241 The tangency portfolio is defined in Equation (4.13).


It contains enormous long and short positions primarily in the 6-, 7- and 8-year zero-coupon bonds.242 According to our solution the investor is supposedto buy 4313 units of account of face value of the 6-year zero-coupon bond forevery unit of account of initial wealth. The reason for the unrealistic portfoliocomposition is the following: A profound inspection of the tangency portfolioreveals that the sign of the position alternates, i.e. a long position in theTi-year zero-coupon bond is followed by a short position in the Ti + 1-yearzero-coupon bond (and vice versa). Furthermore, for adjacent maturities thecorrelations are highest. These zero-coupon bonds are therefore considerednear perfect substitutes (from a diversification perspective) and so (becauseof differences in expected values and standard deviations) the optimizationprocedure tries to exploit a suspected arbitrage opportunity by buying onezero-coupon bond and going short the other.

In order to obtain portfolios that can be implemented in practice, weprohibit short sales.243 A drawback of this measure is the lack of an analyticsolution for the bond portfolio optimization problem. Another option wouldbe to restrict the number of zero-bonds the investor can buy. This is suggestedby Korn/Koziol (2006). They propose to consider only so many bonds as theterm structure model has risk factors.244 They give however no guidance onwhich bond(s) to select, so this methods seems ad-hoc and is not consideredhere.

Table 4.6 contains the portfolio weights for the ten zero-coupon bonds fordifferent expected portfolio values.245

242 The analysis by Korn/Koziol (2006) finds similar portfolios.243 This is a common investment constraint for mutual funds in practice. § 59 InvG

(German Investment Act) forbids for example short sales for German mutualfunds.

244 See Korn/Koziol (2006), p. 22.245 The gray column indicates the position in the riskless zero-coupon bond. The con-

strained optimization problem was solved with Wolfram Research’s Mathematica5.2 package using the NMinimize function.


Table 4.6. Vasicek model: Zero-coupon bond weights for short-sale constrainedportfolios.

E[W1] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.028 1 0 0 0 0 0 0 0 0 0

1.029 0.44 0.56 0 0 0 0 0 0 0 0

1.031 0 0.85 0.15 0 0 0 0 0 0 0

1.032 0 0.18 0.82 0 0 0 0 0 0 0

1.034 0 0 0.45 0.55 0 0 0 0 0 0

1.035 0 0 0 0.58 0.42 0 0 0 0 0

1.037 0 0 0 0 0.59 0.41 0 0 0 0

1.038 0 0 0 0 0 0.41 0.59 0 0 0

1.040 0 0 0 0 0 0 0.37 0.14 0.49 0

1.042 0 0 0 0 0 0 0 0 0 1

These mean-variance efficient portfolios contain at most three zero-couponbonds. This is not surprising since all assets are nearly perfectly correlated.Any given expected value can therefore be obtained by a linear combination ofjust two assets since adding more assets wouldn’t diversify the portfolio muchmore.246 Nevertheless, it is interesting to note, that the portfolios consist ofdifferent bonds and not only positions in the long (maximum maturity) andshort (minimum maturity) bond for example.

Table 4.7 compares the standard deviations of terminal wealth for the un-constrained and the short-sale constrained case. Although the portfolio com-position differs significantly, the difference in the portfolio standard deviationfor the same expected terminal wealth is small.

Table 4.7. Vasicek model: Standard deviations of terminal wealth for unconstrainedand short-sale constrained portfolios

E[W1] unconstrained constrained

1.028 0.0000 0.00001.029 0.0075 0.00761.031 0.0149 0.01511.032 0.0224 0.02271.034 0.0299 0.03031.035 0.0374 0.03791.037 0.0449 0.04561.038 0.0523 0.05321.040 0.0598 0.06091.042 0.0673 0.0685

246 Recall the two asset Markowitz diagram for varying correlations in classic financetextbooks, e.g. Elton et al. (2003), p. 77


As can be seen from Table 4.7, the loss in “efficiency” (i.e. higher standarddeviation) due to the introduction of short sale constraints is negligible. This isnot surprising since we already mentioned that the assets are nearly perfectlycorrelated and are therefore quite perfect substitutes. Constraining positionsin these assets forces switching but this forced switching is relatively costlessfor the investor, i.e. it raises the portfolio standard deviation only slightly.

Next, we analyze the long-term portfolio selection problem.

Long-term investment horizon

If the investment horizon is greater than one period, then the portfolio value atthe investment horizon is not only determined by the term structure at time Tbut also by the term structures at every date before T , because all cash flowsthat are received before the investment horizon are reinvested at the currentspot rate until T . Changes in the short rate have therefore radically differenteffects on short247 and long bonds248 – analogous to the classical durationanalysis (reinvestment risk versus market risk).249 The major change fromthe short-term to long-term case therefore is the correlation matrix Ψ of therisky zero-coupon bonds250

Ψ =

1 0.76 0.67 0.62 −0.59 −0.59 −0.59 −0.59 −0.590.76 1 0.88 0.81 −0.77 −0.77 −0.77 −0.77 −0.770.67 0.88 1 0.93 −0.88 −0.88 −0.88 −0.88 −0.880.62 0.81 0.93 1 −0.95 −0.95 −0.95 −0.95 −0.95−0.59 −0.77 −0.88 −0.95 1 0.99 0.99 0.99 0.99−0.59 −0.77 −0.88 −0.95 0.99 1 0.99 0.99 0.99−0.59 −0.77 −0.88 −0.95 0.99 0.99 1 0.99 0.99−0.59 −0.77 −0.88 −0.95 0.99 0.99 0.99 1 0.99−0.59 −0.77 −0.88 −0.95 0.99 0.99 0.99 0.99 1

Next, we calculate the tangency portfolio as in the last section. For aninitial wealth of W0 = 1 units of account, the tangency portfolio Ntan is

Ntan =

0000

8.14−22.7031.51−21.745.96

(4.22)

247 Maturity less than T .248 Maturity greater than T .249 See Garbade (1996), pp. 37–40.250 The correlation matrix is truncated to two decimal places for presentational pur-

poses.


In comparison to the short-term case251, this tangency portfolio still containssubstantial long and short positions. For longer maturity bonds, the alter-nating signs that we discovered in last section’s tangency portfolio are alsovisible. This is not surprising since as can be seen from the above correlationmatrix, long term bonds are highly correlated among each other. We pick upthe argumentation from the short-term case and restrict short sales in orderto obtain portfolios that could be implemented in practice.

The mean-variance efficient portfolios with short-sale constraints are shownin Table 4.8.252

Table 4.8. Vasicek model: Zero-coupon bond weights for short-sale constrainedportfolios.

E[W5] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.168 0 0 0 0 1 0 0 0 0 0

1.172 0.02 0.03 0.06 0.14 0.08 0.66 0 0 0 0

1.176 0.04 0.07 0.14 0.09 0 0.03 0.63 0 0 0

1.181 0.06 0.11 0.19 0 0 0 0 0.64 0 0

1.185 0.07 0.17 0.12 0 0 0 0 0 0.64 0

1.189 0.08 0.22 0.04 0 0 0 0 0 0 0.65

1.194 0.08 0.19 0 0 0 0 0 0 0 0.74

1.198 0.06 0.11 0 0 0 0 0 0 0 0.83

1.202 0.05 0.04 0 0 0 0 0 0 0 0.91

1.207 0 0 0 0 0 0 0 0 0 1

In contrast to the short-term case considered in the last section, the in-vestor holds more zero-coupon bonds (up to six). The structure of the bondportfolios differ also. In the short-term case, the maturity of the zero-couponbonds was concentrated at one point on the maturity spectrum.253 The long-term case produces efficient portfolio that consist of short and long bonds butno positions in intermediate bonds.

Next, we compare the short-sale constrained optimization to the uncon-strained optimization in terms of differences in portfolio standard deviationfor specific expected values of terminal wealth.

251 See Equation (4.21).252 The gray column indicates the position in the riskless zero-coupon bond. The con-

strained optimization problem was solved with Wolfram Research’s Mathematica5.2 package using the NMinimize function.

253 For a given expected value of terminal wealth.


Table 4.9. Vasicek model: Standard deviations of terminal wealth for unconstrainedand short-sale constrained portfolios


1.16775 0.0000 0.00001.17208 0.0095 0.01011.17641 0.0193 0.02041.18075 0.0290 0.03091.18508 0.0386 0.04161.18941 0.0483 0.05241.19374 0.0579 0.06341.19808 0.0676 0.07481.20241 0.0773 0.08621.20674 0.0869 0.0978

The result of the comparison is quite the same as in the short-term invest-ment horizon case. The rise in portfolio standard deviation due to short-saleconstraints is negligible, although the portfolio compositions vary significantly.

In this section we have shown how the mean-variance approach can beused for bond portfolio selection. The bond market was governed by the Va-sicek term structure model. An unconstrained optimization yielded portfolioswith huge long and short positions. Since these can’t be implemented in thereal world, we constrained short sales. The resulting portfolios were muchmore plausible. Another interesting point is that the introduction of shortsale constraints didn’t have a significant influence on the portfolio standarddeviations. In the next section we analyze the results of the HW2 model.

4.2.5 Two-Factor Hull/White (1994) Model

We derived the HW2 model in Chapter 3.5. It assumes two bivariate normallydistributed state variables, r(t) and ε(t). The bond portfolio selection param-eters can then be calculated easily since this model is a special case of themulti-factor Gaussian setting outlined in Chapter 4.2.2 with x(t)′ = (r(t), ε(t))and

B(t, T )′ = (B1(t, T ), B2(t, T ))

The covariance matrices are then defined as follows:

COV(x(t), x(t)′) =(

var(r(t)) cov(r(t), ε(t))cov(ε(t), r(t)) var(ε(t))

)

and

COV(x(t), x(τ)′) =(

cov(r(t), r(τ)) cov(r(t), ε(τ))cov(ε(t), r(τ)) cov(ε(t), ε(τ))

)

Hull/White (1996) argue that, when considering a two-factor model, usinga simple best-fit calibration technique to match the current term structure of


interest rates is not the right choice, instead one should set some parametersto values that make economic sense and only fit the remaining parameters tothe current term structure of interest rates.254

We adopt this approach and assume economically plausible values for thecurrent spot rate r(0) and for the correlation coefficient %.255 Table 4.10 sum-marizes the parameter values

Table 4.10. HW2 model: Parameter values for numerical example.

Parameter Value

r(0) 0.025ε(0) 0% 0.6θ 0.0053κr 0.2591κε 0.8274σr 0.0073σε 0.0219λ1 1.2395λ2 0

This choice of parameters approximates the German term structure ofinterest rates as of January 2006 quite good and furthermore results in aplausible correlation structure.256

Short-term investment horizon

First, we want to examine the case T = 1. The major difference between one-factor and two-factor models is the correlation structure between spot interestrates of different maturity.257

Given the above parameters, the HW2 model yields the following correla-tion matrix for the zero-coupon bonds.258

254 See Hull/White (1996), p. 289.255 The HW2 model assumes ε(0) = 0.256 As shown in Equation (4.23).257 In a one factor affine model, all correlations are equal to 1.258 As usual it is truncated for presentational purposes to two decimal places.


Ψ =

1 0.9 0.72 0.58 0.48 0.42 0.38 0.35 0.330.9 1 0.95 0.87 0.81 0.77 0.74 0.72 0.710.72 0.95 1 0.98 0.96 0.93 0.91 0.9 0.890.58 0.87 0.98 1 0.99 0.98 0.97 0.97 0.960.48 0.81 0.96 0.99 1 0.99 0.99 0.99 0.990.42 0.77 0.93 0.98 0.99 1 0.99 0.99 0.990.38 0.74 0.91 0.97 0.99 0.99 1 0.99 0.990.35 0.72 0.9 0.97 0.99 0.99 0.99 1 0.990.33 0.71 0.89 0.96 0.99 0.99 0.99 0.99 1

(4.23)

The correlation matrix a two-factor model captures the real-world correla-tions of the bond market quite well. Zero-coupon bond prices of different matu-rities are positively but not perfectly correlated. The higher the maturity dif-ference between zero-coupon bonds, the lower is the correlation between them,e.g. corr(P (1, 2), P (1, 3)) = 0.9 and corr(P (1, 2), P (1, 10)) = 0.33. But for afixed maturity difference (e.g. 1 period) the correlations get higher the morethe maturity differs from the investment horizon, e.g. corr(P (1, 2), P (1, 3)) =0.9 but corr(P (1, 6), P (1, 7)) ≈ 1.

Figures 4.3 and 4.4 show the current (and expected) term structures aswell as the term structure of volatility.

2 4 6 8 10T

0.0225

0.025

0.0275

0.03

0.0325

0.035

0.0375

RH0,TL

Fig. 4.3. HW2: Term structure

2 4 6 8T

0.002

0.004

0.006

0.008

0.01

0.012

0.014

ΣRH0,TL

Fig. 4.4. HW2: Volatility structure

First we derive the expected returns of zero-coupon bonds of differentmaturities. These are shown in Table 4.11. We observe that an investment inlonger term zero-coupon bonds promises a higher expected return. This is notsurprising since interest rates are expected to decline259and therefore longerterm bonds are expected to gain more.

Table 4.11. HW2 model: Expected holding period returns of zero-coupon bonds.

Ti 1 2 3 4 5 6 7 8 9 10

Exp. ret. (%) 2.681 3.071 3.354 3.553 3.69 3.785 3.852 3.899 3.934 3.958

259 More precisely a nearly parallel downward movement for the relevant maturities.See Figure 4.3.


In order to derive mean-variance efficient portfolios, we calculate the tan-gency portfolio. It contains – as has been the case in the Vasicek model – verylarge long and short positions.

Ntan =

28.98−325.551840.40−5314.765932.306187.92−24694.2025114.00−8781.28

(4.24)

Since constructing such a portfolio is highly unrealistic in practice, we con-centrate once again on the short-sale restricted case. We give the portfolioweights for different expected values of terminal wealth in Table 4.12.260

Table 4.12. HW2 model: Zero-coupon bond weights for short-sale constrained port-folios.

E[W1] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.027 1 0 0 0 0 0 0 0 0 0

1.029 0.76 0.08 0.12 0.04 0 0 0 0 0 0

1.030 0.56 0 0.44 0 0 0 0 0 0 0

1.032 0.25 0.36 0.23 0.18 0 0 0 0 0 0

1.033 0.12 0 0.88 0 0 0 0 0 0 0

1.035 0 0 0.67 0.33 0 0 0 0 0 0

1.036 0 0 0 0.91 0.09 0 0 0 0 0

1.038 0 0 0 0 0.86 0.14 0 0 0 0

1.040 0 0 0 0 0 0.24 0.76 0 0 0

1.041 0 0 0 0 0 0 0 0 0 1

For lower expected values of terminal wealth, the portfolios are quite diver-sified with up to four zero-coupon bonds contained in the portfolio. For higherexpected values, the portfolios contain at most two zero-coupon bonds. Di-versification is apparently only possible for lower expected values. A furtherlook at the correlation matrix clarifies this result. Long-term bonds are highlycorrelated among each other and these are the bonds that offer the higher ex-pected returns necessary for high expected terminal wealth. In order to achievehigh expected returns, the investor has to buy long-term bonds but then thediversification potential vanishes because these bonds are highly correlated

260 The gray column indicates the riskless zero-coupon bond. The constrained opti-mization problem was solved with Wolfram Research’s Mathematica 5.2 packageusing the NMinimize function.


and therefore only two bonds are included in the optimal portfolio. For lowerexpected portfolio returns, short and long bonds can be bought and betweenthese “groups”, the correlation is quite low. The structure of the portfoliosresembles the short-term Vasicek case, because normally no position is takenin extreme maturity sectors.

Next, we again examine the loss in diversification due to short-sale con-straints. Table 4.13 contains the portfolio standard deviations for the uncon-strained and the short-sale constrained optimal portfolios for different ex-pected values.

Table 4.13. HW2 model: Standard deviations of terminal wealth for unconstrainedand short-sale constrained portfolios


1.027 0.0000 0.00001.029 0.0005 0.00281.030 0.0010 0.00561.032 0.0016 0.00851.033 0.0021 0.01131.035 0.0026 0.01431.036 0.0031 0.01821.038 0.0037 0.02341.040 0.0042 0.03021.041 0.0047 0.0386

It can be seen from this table that in the HW2 model, when consideringa short-term investment horizon, there are sizeable increases in portfolio riskdue to the introduction of short-sale constraints.

Long-term investment horizon

For a long-term investment horizon (T = 5) the results turn out to be nearlyidentical to the short term case although some correlations are now nega-tive.261 The unconstrained portfolios contain again large long and short po-sitions, so we will give the results for the short-sale constrained optimizationonly. Table 4.14 gives the portfolio weights for the assets in the short-saleconstrained case.262

261 This has also been the case in the long-term Vasicek model.262 The gray column indicates the riskless zero-coupon bond. The constrained opti-

mization problem was solved with Wolfram Research’s Mathematica 5.2 packageusing the NMinimize function.


Table 4.14. HW2 model: Zero-coupon bond weights for short-sale constrained port-folios.

E[W5] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.169 0 0 0 0 1 0 0 0 0 0

1.174 0 0 0 0 0.85 0 0.01 0.14 0 0

1.178 0 0 0 0 0.71 0 0.02 0.27 0 0

1.183 0 0 0 0 0.56 0 0.03 0.41 0 0

1.187 0 0 0 0 0.41 0 0.04 0.55 0 0

1.192 0 0 0 0 0.26 0 0.05 0.69 0 0

1.196 0 0 0 0 0.11 0 0.07 0.82 0 0

1.200 0 0 0 0 0 0 0 0.92 0.08 0

1.205 0 0 0 0 0 0 0 0.09 0.91 0

1.209 0 0 0 0 0 0 0 0 0 1

It is interesting to note that no positions are held in short bonds, i.e.no position is taken in bonds with a maturity shorter than 5 years. In thisexample – from a mean-variance perspective – rolling over, i.e. investing inshort bonds and reinvesting until the investment horizon, seems to be aninefficient strategy. Furthermore, it is conspicuous that quite large positionsare held in the riskless bond.263 Regarding the structure of the portfolios, theoutcome resembles the short-term case, i.e. the resulting portfolios resemblebullet portfolios.

Table 4.15 summarizes the mean-variance information of the constrainedand unconstrained portfolios.

Table 4.15. HW2 model: Standard deviations of terminal wealth for unconstrainedand short-sale constrained portfolios


1.169 0.0000 0.00001.174 0.0032 0.01051.178 0.0065 0.02101.183 0.0097 0.03141.187 0.0130 0.04191.192 0.0162 0.05241.196 0.0194 0.06291.200 0.0227 0.07341.205 0.0259 0.08441.209 0.0292 0.0961

As can be seen from Table 4.15, there are sizeable increases in portfoliorisk due to the introduction of short-sale constraints.

263 This is in contrast to the findings for the Vasicek model. There, nearly no positionswere taken in the riskless bond, see Table 4.8.


In this section the bond market was governed by the HW2 term structuremodel. An unconstrained optimization yielded portfolios with large long andshort positions. Hence, we again constrained short sales. The resulting port-folios were much more plausible. In contrast to the findings in the Vasicekmodel, the introduction of short sale constraints had a significant effect onthe portfolio standard deviations.

In the next part, we want to analyze how mean-variance efficient port-folios compare to portfolios resulting from active and passive bond portfoliostrategies employed in practice in various interest rate scenarios.

4.3 Static Bond Portfolio Selection in Practice

4.3.1 Introduction

Despite its broad acceptance in the equity markets (by both academics andpractitioners alike), the mean-variance framework could never quite establisha foothold in the fixed-income markets.264 This might be due to the addedcomplexity as has become apparent from the analysis in the last section or asWilhelm (1992) suggests, due to low interest rate volatility in the 1970s.265

In this chapter we want to compare common bond portfolio selection tech-niques used in practice with mean-variance efficient portfolios resulting fromthe framework introduced in the last section by means of numerical exam-ples.266 Since there exist numerous bond portfolio selection methods in prac-tice, we have to select methods that are representative and lend itself easilyto a comparison with the mean-variance approach.

Bond portfolio selection strategies can be classified in various ways. A com-mon approach is to distinguish between active and passive portfolio selectionstrategies.267 An active strategy can be defined as an investment strategy thatrequires substantial expectational input, i.e. in our context, the investor musthave a “view” on the term structure evolution.268 Passive strategies on theother hand require no or only minimal expectational input.269 In the course ofthis section we will examine both strategies. We analyze active bond portfolioselection strategies first.

264 The classic book by Fabozzi (2004) doesn’t even mention mean-variance analysisas a possible portfolio selection tool.

265 See Wilhelm (1992), p. 210.266 For a complete treatise on bond portfolio selection see e.g. Fabozzi (2000), Fabozzi

(2001), Fabozzi (2004) and Martellini/Priaulet/Priaulet (2003).267 See Martellini/Priaulet/Priaulet (2003), p. 211.268 See Fabozzi (2004), p. 412.269 See Fabozzi (2004), p. 412.

4.3 Static Bond Portfolio Selection in Practice 67

4.3.2 Active Bond Portfolio Selection Strategies

The practice generally distinguishes at least three active bond portfolio selec-tion strategies:270

• Riding/rollover strategies• Duration strategies and• Yield curve strategies

Riding the Yield Curve and Rollover Strategies

Riding the yield curve entails buying longer dated bonds271 and selling thembefore maturity.272 Rollover refers to the strategy of buying shorter datedbonds, holding them until maturity and investing the proceeds again.273 Bothstrategies rely on the view that the current term structure of interest ratesremains relatively stable.274

According to the practice, the selection of the strategy (riding vs. rollover)depends on the slope of the yield curve.275 When the term structure does notchange276, then every zero-coupon bond with the maturity Ti has a ∆t-holdingperiod return equal to the forward rate between Ti−∆t and Ti.277 The forwardrates are at the same time so-called break-even rates. They show how muchrates can increase before the investor earns less than the riskless spot rate.278

In a positive term structure environment, investors hence utilize the ridingstrategy, because then the holding period return of the longer dated bonds isgreater simply because the maturity shortens.279 On the other hand, if theterm structure is downward sloping, then the investor can gain higher returnsif he employs the rollover strategy. Since the yield curve is upward slopingmost of the time, we concentrate on the riding strategy.

The empirical results on the profitability of the riding strategy are mixed.280

A recent study by Bieri/Chincarini (2005) examines which signal – e.g. pos-itive slope of yield curve – should be used to initiate a riding strategy. Theyfind that investors historically could have significantly enhanced their returnsby riding the yield curve instead of buying and holding zero-coupon bondswith a maturity equal to their investment horizon.281 From a mean-variance

270 See Martellini/Priaulet/Priaulet (2003), pp. 234–240.271 Maturity longer than the investment horizon.272 See Bieri/Chincarini (2005), p. 6.273 See Martellini/Priaulet/Priaulet (2003), p. 237.274 See Martellini/Priaulet/Priaulet (2003), p. 234.275 See Martellini/Priaulet/Priaulet (2003), p. 234.276 R(t, T ) = R(t + ∆T, T + ∆t) ∀ T277 See Ilmanen (1995), p. 4.278 See Bieri/Chincarini (2005), p. 9.279 See Zimmerer (2003), p. 243.280 For a summary of their findings see Martellini/Priaulet/Priaulet (2003), p. 235.281 See Bieri/Chincarini (2005), p. 28.


perspective one could argue that the deviation of the asset maturities fromthe investment horizon in the riding strategy makes the strategy riskier, soone can expect to earn a higher return from taking on the risk. Furthermore,the strategy doesn’t give advice on the bond282 with which to implement thestrategy. A naive approach would be to buy the zero-coupon bond with thebiggest expected holding period return. It becomes obvious that the conceptof diversification is not part of the strategy. If the investor thinks that ridingis preferable to buying and holding, he invests all his money in the respectivebond.

We now compare – by means of a numerical example – the riding strategywith the mean-variance framework. In the Vasicek model, the expected futureterm structure depends only on the short-rate, so when the expected futureshort-rate is the same as today, the whole term-structure is expected to staythe same. As we pointed out in Chapter 3.4 this is the case when r(0) = θ,i.e. the current short rate is equal to the mean-reversion level. We assume aninvestment horizon of T = 1 and an investment universe consisting of τ = 10zero-coupon bonds of different maturities 1, . . . , τ . Furthermore, we assumethe following parameters for the Vasicek term structure model

Table 4.16. Vasicek model: Riding strategy parameter values.

Parameter Value

r(0) 0.024θ 0.024κ 0.1668σr 0.0153λ 0.2126

The term structure of interest rates is then positively sloped283 and ex-pected to remain unchanged, because r(t) = θ. Table 4.17 contains the port-folio weights for mean-variance efficient bond portfolios for different expectedvalues of terminal wealth.

282 It is always a single bond that is bought.283 Because r(t) ≤ R(∞)− σ2

r4κ2 , see Vasicek (1977), p. 168.


Table 4.17. Vasicek model: Portfolio weights when term structure is expected toremain unchanged.

E[W1] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.026 1 0 0 0 0 0 0 0 0 0

1.027 0.44 0.56 0 0 0 0 0 0 0 0

1.029 0 0.85 0.15 0 0 0 0 0 0 0

1.030 0 0.18 0.82 0 0 0 0 0 0 0

1.032 0 0 0.43 0.57 0 0 0 0 0 0

1.034 0 0 0 0.58 0.42 0 0 0 0 0

1.035 0 0 0 0.1 0.38 0.52 0 0 0 0

1.037 0 0 0 0 0 0.53 0.33 0.14 0 0

1.038 0 0 0 0 0 0 0 0.95 0.05 0

1.040 0 0 0 0 0 0 0 0 0 1

The highlighted row in Table 4.17 refers to the buy-and-hold strategy. Allother rows contain the portfolio weights of short-sale restricted mean-varianceefficient portfolios. It is interesting that the optimum portfolios are close tothe ones proposed by the riding strategy.284 The portfolio weights are con-centrated at specific points on the maturity spectrum. But the concentrationis not perfect, i.e. the investor should not buy one single zero-coupon bondbut at least two.285 But – as has been noted before – this being a one-factormodel, the diversification potential is quite limited. So, in practice the differ-ence in terms of portfolio volatility between buying a single zero-coupon bondor a portfolio of bonds with the same expected value is negligible.286 Sincethe results of this analysis are very similar to the numerical example in thelast section, we do not analyze the HW2 model.

The mean-variance framework generates portfolios that are comparable tothose recommended by the riding strategy. It offers furthermore the advantagethat the investor can assess the riskiness of the different possible implemen-tations (zero-coupon bonds to buy) of the strategy better.

Future research could analyze how the riding strategy performs in com-parison to mean-variance efficient portfolios in different scenarios (also whenthe expectations are not met). We suspect that mean-variance efficient portfo-lios would turn out to be superior, because in this framework the uncertaintyregarding the outcome is already being accounted for.

Duration strategies

Naive duration strategies require only expectations about one variable, thelevel of interest rates. They assume, that only one factor drives the term

284 Longer dated bonds are bought.285 This result is not due to the specific expected values chosen, another run with 50

different expected portfolio values resulted in similarly diversified portfolios.286 See Chapter 4.2.4.


structure and this factor affects all rates in the same way, i.e. the term struc-ture is affected only by parallel movements.287

It is well known that – inside the duration framework – the investor canimplement his view about level changes by setting the portfolio’s Macaulayduration relative to his investment horizon. If he expects interest rates to fall(increase), he chooses a portfolio with a duration longer (shorter) than theinvestment horizon. A comparison of the duration strategies and the mean-variance framework is not perfect since – at least in the Vasicek model –there can be no parallel movements of the term structure of interest rates.288

Longer-term rates change less than shorter-term interest rates.289 We nowwant to present two numerical examples and compare the outcomes to theadvice given by the duration strategy. We have to focus on the long term case(investment horizon is 5 years) since in the short term case portfolio durationssmaller than the investment horizon are impossible when short-sales are notallowed.

Falling Interest Rates. In Chapters 4.2.4 and 4.2.5 we presented a nu-merical example for the Vasicek and the HW2 model, where the interest rateswere expected to fall.290 Hence, the duration strategy would advise to con-struct a portfolio with a duration greater than the investment horizon, i.e.greater than 5. A closer look at Tables 4.8 and 4.14 reveals that the portfoliodurations are in all cases291 greater than the investment horizon. The result ofthe (short-sale constrained) mean-variance optimization is hence in line withthe practical duration advice.

Rising Interest Rates. Next we consider the case where the investorexpects interest rates to rise. We adapt the parameters from Table 4.16 inorder to obtain an expected rise in the term structure of interest rates. Weassume the following parameters for the Vasicek model

Table 4.18. Vasicek model: Duration strategy parameter values.

Parameter Value

r(0) 0.02θ 0.024κ 0.1668σr 0.0153λ 0.2126

287 See Martellini/Priaulet/Priaulet (2003), p. 236.288 See Equation (3.32) and Table 3.1. In the Ho/Lee (1986) model however, only

parallel term structure movements can occur.289 In the Vasicek model the infinitely long rate stays constant.290 See Figures 4.1 and 4.3.291 Disregarding the riskless investment option.


The current term structure of interest rates is hence upward-sloping (solidline) and is expected to rise (dashed line) as can be seen from Figure 4.5

2 4 6 8 10T0.018

0.02

0.022

0.024

0.026

0.028

0.03

0.032

RH0,TLE@RH5,T+5LD

Fig. 4.5. Vasicek model: Expected term structure

The duration strategy would advise to construct a portfolio with a durationsmaller than the investment horizon. The portfolio optimization results aregiven in Table 4.19.

Table 4.19. Vasicek model: Portfolio weights and durations for duration strategycomparison.

E[W5] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 MacDur.

1.145 0 0 0 0 1 0 0 0 0 0 5.001.149 0.02 0.03 0.06 0.14 0.08 0.66 0 0 0 0 5.211.153 0.04 0.07 0.15 0.04 0 0.17 0.53 0 0 0 5.521.157 0.06 0.11 0.19 0 0 0 0 0.64 0 0 5.951.162 0.07 0.17 0.12 0 0 0 0 0 0.64 0 6.531.166 0.08 0.22 0.04 0 0 0 0 0 0 0.65 7.181.170 0.08 0.18 0 0 0 0 0 0 0 0.74 7.841.174 0.06 0.11 0 0 0 0 0 0 0 0.83 8.541.179 0.05 0.04 0 0 0 0 0 0 0 0.91 9.241.183 0 0 0 0 0 0 0 0 0 1 10.00

Surprisingly the portfolio (Macaulay) duration is in all relevant casesgreater than the investment horizon, therefore contradicting the advice givenby duration strategy. This result can be reproduced in a similar term structureenvironment with the HW2 model. We assume the following parameters:


Table 4.20. HW2 model: Duration strategy parameter values.

Parameter Value

r(0) 0.018ε(0) 0% 0.6θ 0.0053κr 0.2591κε 0.8274σr 0.0073σε 0.0219λ1 1.2395λ2 0

Table 4.21 summarizes the portfolio weights and the Macaulay portfoliodurations for different expected values of terminal wealth

Table 4.21. HW2 model: Portfolio weights and Macaulay durations for durationstrategy comparison.

E[W5] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 MacDur.

1.146 0 0 0 0 1 0 0 0 0 0 5.00

1.151 0 0 0 0 0.85 0 0 0.15 0 0 5.36

1.155 0 0 0 0 0.71 0 0.02 0.28 0 0 5.86

1.160 0 0 0 0 0.56 0 0.03 0.41 0 0 6.30

1.164 0 0 0 0 0.41 0 0.04 0.55 0 0 6.73

1.168 0 0 0 0 0.26 0 0.05 0.69 0 0 7.16

1.173 0 0 0 0 0.12 0 0.06 0.83 0 0 7.60

1.177 0 0 0 0 0 0 0 0.92 0.08 0 8.08

1.182 0 0 0 0 0 0 0 0.09 0.91 0 8.91

1.186 0 0 0 0 0 0 0 0 0 1 10.00

Both models give hence the same result. A possible explanation is thatin both models, short term rates are more volatile than long term rates, i.e.rolling over is a risky strategy.292 Furthermore, in these examples the expectedvalue of rolling over once293 is not very different from investing in longer termbonds, so rollover strategies seem to be disadvantaged. But at least in theVasicek model, rollover structures are visible in the portfolios.294 In the HW2model on the other side no such structures are visible, the optimum portfolioconsists only of bonds with a maturity of at least 5 years. Another important

292 So less weight is given to short bonds.293 The model is constructed in such a way that there is only one roll-over, the

proceeds from maturing bonds is invested until the investment horizon.294 Zero-coupon bonds with maturities less than T are bought.


point is that there are no parallel movements in the Vasicek model. Wheninterest rates are expected to rise, the change in the 10-year spot rate issmaller than the change in the 2-year spot rate, so the expected capital losson long bonds is relatively smaller than in the duration framework.

Furthermore, we observe that the overall structure of the portfolios isnearly the same regardless of the interest rate scenario considered.295 Thismight be due to the fact, that the expected rise in interest rates is quitesmall.

In summary, we can conclude that the classical duration advice cannot bereproduced in a rising interest rate scenario. This might point to the missingconsideration of risk inside the duration framework or to oversimplification(parallel yield-curve shifts).

Yield Curve Strategies

Yield curve strategies involve positioning a portfolio to capitalize on expectedchanges in the shape of the (risk-free) yield curve.296 Usually one distinguishesthree yield curve strategies: bullet, barbell and ladder strategies.297 In a bulletstrategy, the portfolio is constructed in such a way that the maturity of thesecurities in the portfolio are highly concentrated at one (intermediate) pointon the yield curve.298

1 2 3 4 5 6 7 8 9 10T

0.2

0.4

0.6

0.8

1

wT

Fig. 4.6. Archetypical bullet portfolio

A barbell portfolio is constructed by concentrating investments at theshort-term and the long-term ends of the yield curve.299 A barbell is definedrelative to a bullet strategy, it consists of maturities smaller and greater thanthe bullet maturity.300

295 Compare Tables 4.8 and 4.19 for the Vasicek model and 4.14 and 4.21 for theHW2 model.

296 Fabozzi (2004), p. 424.297 Fabozzi (2004), p. 427.298 Fabozzi (2004), p. 427.299 Martellini/Priaulet/Priaulet (2003), p. 239.300 Fabozzi (2004), p. 427.


1 2 3 4 5 6 7 8 9 10T

0.2

0.4

0.6

0.8

1wT

Fig. 4.7. Archetypical barbell portfolio

In a ladder strategy, the portfolio is constructed to have approximatelyequal amounts of each maturity.301

1 2 3 4 5 6 7 8 9 10T

0.2

0.4

0.6

0.8

1wT

Fig. 4.8. Archetypical ladder portfolio

The fixed income literature dealing with yield curve strategies examinedthe relative performance of these strategies in different yield curve environ-ments.302 The main result is that portfolios with equal duration can havequite different risk profiles, i.e. non-parallel changes in the shape of the yieldcurve can have significant effects on the portfolio return.

Studies showed that a particular yield curve strategy is optimal for a par-ticular interest rate scenario. A flatter yield curve generally303 results in anoutperformance of the barbell strategy (given a downward or upward move-ment of yield curve) and a steeper yield curve results in an outperformance ofthe bullet strategy.304 Fabozzi (2004) finds that for very large level changes305,a steepening yield curve leads to an outperformance of the barbell strategy –therefore contradicting Jones (1991).306 Mann/Ramanlal (1997) find that forlonger maturities the “normal results” of Jones (1991) are reversed.307

301 Fabozzi (2004), p. 427.302 See e.g. Jones (1991), Willner (1996) and Mann/Ramanlal (1997).303 The exceptions are considered below.304 See Jones (1991), p. 48.305 More than ±300bp306 See Fabozzi (2004), p. 431.307 See Mann/Ramanlal (1997), p. 69.


Next, we want to examine whether bullet, barbell or ladder portfolios resultfrom our mean-variance optimization framework. First, we consider the caseof an expected steepening of the term structure and thereafter the case of anexpected flattening.

Steepening Term Structure. First, we have a look at the Vasicek model.As we have reiterated in the last section, there are no parallel term structuremovements in the Vasicek model. A rising short rate moves the whole termstructure upwards (except the infinitely long rate), so the yield spread308 be-comes smaller, i.e. the term structure flattens. On the other hand, a decreasingshort rate moves the whole term structure downwards and results in a steepercurve. These two kinds of term structure movements are historically the mostcommon as has been found by Jones (1991). Yield curve strategies are tradi-tionally examined for short-term investment horizons, i.e. in our frameworkfor an investment horizon of 1 year.309

We already presented numerical examples for both the Vasicek and theHW2 model, that were set in such an environment. The Vasicek numericalexample for the short term investment horizon case that we considered inChapter 4.2.4, was set in a positive yield curve environment and the termstructure was expected to decrease and hence to steepen. The result we ob-tained310 was in accordance with the practical advice from yield curve strate-gies. We expected the term structure to steepen, an environment that favorsbullet portfolios.

In the HW2 numerical example in Chapter 4.2.5, the term structure ofinterest rates was expected to steepen and therefore – according to the practice– bullet portfolios should outperform barbell portfolios. The mean-varianceefficient portfolios given in Table 4.12 can be regarded as bullet portfolios311,therefore the theoretical model comes to the same conclusion as the practicaladvice for the steepening case.

Flattening Term Structure. The numerical example for the Vasicekmodel that we presented in Chapter 4.3.2 was set in a positive yield curveenvironment with an expected rise in the term structure. But the results weobtained were for the long-term investment horizon case (5 years). We usethe same parameters and calculate the optimum portfolios for the short-termcase. Let r(0) = 0.02, then the term structure is expected to increase andtherefore to flatten.312 In such an environment, the practice would propose abarbell portfolio. But as is shown in Table 4.22 the mean-variance approachresults in bullet portfolios.

308 Normally defined as the difference between the 10 year spot rate and the 2 yearspot rate, we define the yield spread as the difference between the infinitely longrate and the short rate.

309 See Fabozzi (2004), p. 427.310 See Table 4.6.311 The investment is usually concentrated among bonds with adjacent maturities.312 The other parameters of the term structure model remain the same as in Chapter

4.3.2.


Table 4.22. Vasicek model: Zero-coupon bond weights when a flattening is ex-pected.

E[W1] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.022 1 0 0 0 0 0 0 0 0 0

1.024 0.44 0.56 0 0 0 0 0 0 0 0

1.025 0 0.85 0.15 0 0 0 0 0 0 0

1.027 0 0.18 0.82 0 0 0 0 0 0 0

1.028 0 0 0.43 0.57 0 0 0 0 0 0

1.030 0 0 0.09 0.38 0.53 0 0 0 0 0

1.031 0 0 0 0 0.59 0.41 0 0 0 0

1.033 0 0 0 0 0 0.41 0.59 0 0 0

1.034 0 0 0 0 0 0 0 0.95 0.05 0

1.036 0 0 0 0 0 0 0 0 0 1

Next, we consider the case of a flattening term structure of interest inthe HW2 model.313 Let r(0) = 0.015 and keep the other parameters theunchanged314, then the term structure of interest rates is expected to increaseand to flatten. But as is shown in Table 4.23 the mean-variance approachagain results in bullet portfolios.315

Table 4.23. HW2 model: Zero-coupon bond weights when a flattening is expected.

E[W1] w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

1.018 1 0 0 0 0 0 0 0 0 0

1.020 0.78 0 0.22 0 0 0 0 0 0 0

1.021 0.49 0.27 0.09 0.15 0 0 0 0 0 0

1.023 0.34 0 0.66 0 0 0 0 0 0 0

1.024 0.12 0 0.88 0 0 0 0 0 0 0

1.026 0 0 0.67 0.33 0 0 0 0 0 0

1.027 0 0 0 0.93 0.05 0.02 0 0 0 0

1.029 0 0 0 0 0.86 0.14 0 0 0 0

1.030 0 0 0 0 0 0.24 0.76 0 0 0

1.032 0 0 0 0 0 0 0 0 0 1

In our numerical examples, it seems that regardless of the expected twistin the term structure (steepening or flattening) bullet portfolios are mean-variance efficient when short-sales are not allowed.316 A reason for this might

313 We cannot use the results from Table 4.21 since this example was set with long-term investment horizon environment.

314 See page 71.315 The initial wealth is spread among bonds with adjacent maturities.316 Compare Tables 4.6 and 4.22 for the Vasicek and Tables 4.12 and 4.23 for the

HW2 model.


be, that in these term structure models a twist in the term structure (flatteningor steepening) can’t occur independently of a change in the overall level.Hence, two effects occur at the same time, a change in the level of the termstructure and a change in the shape. Nevertheless, the practical advice tobuy barbell portfolios when the term structure of interest rates is expected toflatten can’t be replicated in our numerical examples.

4.3.3 Passive Bond Portfolio Selection Strategies

Passive portfolio selection is a strategy that requires no – or only minimal –expectational input; typical passive portfolio strategies include indexing andimmunization.317

Immunization Strategy

The immunization strategy318 tries to construct a bond portfolio in such a waythat the portfolio value at the investment horizon of the investor is immuneto a change in interest rates. It is well known that, under the assumptionsthat the term structure of interest rates is flat and that it changes only ina parallel fashion, this is achieved by constructing a bond portfolio with aduration equal to the investment horizon of the investor.319 Disregarding theno-short-sales constraint, every possible portfolio duration can be obtained bypositions in only two bonds.320 In this framework, the investor is indifferentbetween two portfolio with the same duration.

In this section, we compare minimum-variance portfolios321 in the mean-variance framework described in Chapter 4.2.2 with the duration-based immu-nization portfolios. The minimum-variance portfolio should – in our opinion– be conceptually as close as it can get to an immunized portfolio. Before wecompare the portfolios, we have to adjust the model. The model presentedin Chapter 4.2.2 assumes the existence of a riskless zero-coupon bond, i.e. azero-bond with a maturity equal to the investment horizon of the investor. Ifwe would include this zero-coupon bond, then the minimum-variance portfoliowould consist only of the riskless zero-coupon bond and would be identicalto the duration immunized portfolio. Hence, for our comparison, we assume

317 See Fabozzi (2004), p. 412.318 Redington (1952) is generally credited with pioneering this strategy.319 See Fabozzi (2004), p. 476. The assumption underlying the duration model – a

flat term structure that shifts in a parallel fashion – are (i) empirically unlikelyand (ii) would represent an arbitrage opportunity. So even if the duration modelcould been applied to bond portfolio selection, the question remains, whethersuch an application is sensible.

320 If short sales aren’t allowed, only portfolio durations between the shortest and thelongest bond duration are possible but these could still be obtained by positionsin just two bonds.

321 Formally defined in Equation (4.25).


that this riskless zero-coupon bond does not exist. We assume, a bond marketwhere zero-coupon bonds of maturities 1, . . . , T − 1, T + 1, . . . , τ trade. Theinvestment horizon of the investor is T with 1 < T < τ . In our numericalexample, we set τ = 10. Hence, there are nine risky zero-coupon bonds toinvest in. For the sake of analytical tractability, we restrict our analysis againto the two dynamic term structure models we presented in Chapters 3.4 and3.5, namely the Vasicek (1977) and the HW2 model.

For each term structure model, we consider investment horizons from 2 to9 years. For each investment horizon, we then calculate the minimum-varianceportfolio (subject to short-sale constraints). For comparison, the duration im-munization portfolio is constructed by identical weights of the T − 1 and theT + 1 zero-coupon bonds.322 Since we disregard the existence of a risklesszero-coupon bond, we must restate the minimum-variance portfolio optimiza-tion problem since it differs from the optimum solution presented in Chapter4.2.2.

The minimum-variance portfolio is the solution to the following optimiza-tion problem

minN

N ′CN (4.25)

s.t. N ′P0 = W0

N ≥ 0

where N is the ((τ − 1)× 1) holdings vector, C is the covariance matrix, P0 isthe current price vector, W0 is the initial wealth and 0 is a τ−1 vector of zeros.Because of the short-sale constraints, this quadratic optimization problem hasno analytical solution.323

We first analyze the results obtained from the Vasicek model. We choosethe same parameters for the Vasicek term structure model as in Chapter4.2.4.324 The minimum-variance portfolios for different investment horizons(from 2 to 9) are then given in Table 4.24.

322 This represents only one possibility of constructing a portfolio with a duration ofT .

323 The numerical calculations are performed with Mathematica 5.2.324 See Table 4.2.


Table 4.24. Vasicek model: Portfolio weights of minimum-variance portfolios undershort-sale constraints for different investment horizons.

T w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 MacDur.

2 0.58 0 0.42 0 0 0 0 0 0 0 1.84

3 0 0.54 0 0.46 0 0 0 0 0 0 2.92

4 0 0 0.53 0 0.47 0 0 0 0 0 3.94

5 0 0 0 0.52 0 0.48 0 0 0 0 4.96

6 0 0 0 0 0.51 0 0.49 0 0 0 5.98

7 0 0 0 0 0 0.51 0 0.49 0 0 6.98

8 0 0 0 0 0 0 0.51 0 0.49 0 7.98

9 0 0 0 0 0 0 0 0.51 0 0.49 8.98

It is interesting to note that for every possible investment horizon T onlythe T − 1 and the T + 1 period zero-coupon bonds are included in the bondportfolio. From a risk minimizing perspective it seems that concentrating thematurities of the zero-coupon bonds around the investment horizon is opti-mal. A look at the correlations confirms this; the chosen zero-coupon bondsare the ones with the minimum correlation of all available assets, e.g. forT = 2 the minimum correlation is corr( 1

P (1,2) , P (2, 3)) = −0.763153. The du-ration immunization portfolios consist of equal weights in the T − 1 and theT + 1 period bonds. Therefore, in general, the minimum-variance portfolioputs more weight on the short-term bond. But for larger time-horizons, thisdifference becomes smaller. Hence, the Macaulay durations of the minimum-variance portfolios are always smaller than the immunized portfolio durations(which are equal to the investment horizon). But, at the end of the day, theportfolio composition is not that important but the mean-variance charac-teristics of the terminal wealth. Table 4.25 gives the standard deviations ofminimum-variance portfolios in the Vasicek model and immunized portfolios,constructed according to the descriptions above.325

Table 4.25. Vasicek model: Comparison of minimum-variance portfolio standarddeviations to duration-immunized portfolios.

T E[WT ] Vasicek std. Duration std.

2 1.03 0.0054 0.00593 1.04 0.0049 0.00514 1.06 0.0044 0.00455 1.08 0.0039 0.00406 1.10 0.0034 0.00357 1.12 0.0030 0.00318 1.14 0.0027 0.00279 1.16 0.0024 0.0024

325 The standard deviation for the immunized portfolio has been calculated by√N ′

dCNd where Nd is the holdings vector of the immunized portfolio.


It can be seen that for short-term investment horizons, the portfolio stan-dard deviation differs more than for long-term investment horizons. This is notsurprising since for long-term investment horizons, the portfolios are nearlyidentical. But from a practical perspective we can conclude that the differencesare negligible and therefore from a mean-variance perspective, minimum-variance portfolios and duration-immunized portfolios are identical in ourexample.

We now consider the HW2 model. The one-factor Vasicek model did pro-duce nearly identical results to the duration model – regarding both portfolioweights and the mean-variance profile of terminal wealth. In hindsight thismight have been expected, because both models have only one source of ran-domness. The term structure of interest rates in the Vasicek model doesn’tshift in a parallel fashion but still all rates increase or decrease at the sametime. The introduction of a second factor results in the possibility of twists(e.g. short-term interest rates increase but long-term interest rates decrease).Therefore a two-factor model may produce significantly different results.

The HW2 minimum-variance portfolios for investment horizons rangingfrom 2 to 9 years are then given in Table 4.26326

Table 4.26. HW2 model: Portfolio weights of minimum-variance portfolios undershort-sale constraints for different investment horizons.

T w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 MacDur.

2 0.72 0 0.28 0 0 0 0 0 0 0 1.56

3 0.21 0.44 0 0.35 0 0 0 0 0 0 2.49

4 0.11 0 0.48 0 0.41 0 0 0 0 0 3.60

5 0.05 0 0 0.50 0 0.45 0 0 0 0 4.75

6 0.03 0 0 0 0.50 0 0.47 0 0 0 5.82

7 0.01 0 0 0 0 0.51 0 0.48 0 0 6.91

8 0 0 0 0 0 0 0.51 0 0.49 0 7.98

9 0 0 0 0 0 0 0 0.51 0 0.49 8.98

In contrast to the one-factor example, the minimum-variance portfoliosnow usually contain three bonds. In addition to the T − 1 and T + 1 periodbond, it includes non-negligible positions327 in the short bond. Hence, theMacaulay portfolio durations of the minimum-variance portfolios differ forshort-term investment horizons significantly from the immunized ones, e.g. the2-period minimum-variance portfolio has a duration of just 1.56 in comparisonto a duration of 2.0 for the immunized portfolio. Next, we have again a lookat the mean-variance profiles in Table 4.27

326 Parameter values for the term structure model are given in Table 4.10.327 At least for short-term investment horizons.


Table 4.27. HW2 model: Comparison of minimum-variance portfolio standard de-viations to duration-immunized portfolios.

T E[WT ] HW2 std. Duration std.

2 1.03 0.0055 0.00743 1.05 0.0068 0.00814 1.06 0.0068 0.00755 1.08 0.0061 0.00656 1.10 0.0052 0.00547 1.12 0.0044 0.00458 1.14 0.0036 0.00379 1.16 0.0030 0.0030

It can be seen that for short-term investment horizons, the portfolio stan-dard deviation differs more than for long-term investment horizons. Again,this is not surprising since for long-term investment horizons, the portfoliosare nearly identical. For shorter investment horizons, there is a non-negligiblegain in standard deviation reduction from holding the minimum-variance port-folio instead of the duration immunization portfolio. For all other investmenthorizons, we can assume that this gain is negligible. But since portfolio immu-nization strategies are carried out mainly for short-term investment horizons,this still is an important result.

We can conclude that in a one-factor term structure model, the differencesbetween these two strategies are insignificant. Our numerical example for theone-factor Vasicek model showed that the portfolio composition is nearly iden-tical and a possible reduction in portfolio variance when holding the minimum-variance portfolio is negligible. Two-factor interest rate models allow for morerealistic term structure movements. In contrast to one-factor models, so-calledtwists (short-term and long-term rates move in different directions) in the termstructure are now possible. But the effect on the immunization performanceof the duration portfolio seems to be small. Only for short-term investmenthorizons is the gain from switching to the minimum-variance portfolio non-negligible. A duration-immunization portfolio hence performs quite well in atwo-factor term structure model. When more factors are added, the immu-nization performance is likely to be poorer.

The comparison outlined above can be linked to studies investigating im-munization risk. Fong/Vasicek (1984) explore the impact of portfolio structureon the immunization performance of duration-matched portfolios. In otherwords, in light of the different ways in which the term structure of interestrates can shift, is it possible to develop a criterion for minimizing the riskthat a duration-matched portfolio will not be immunized?328 We found thatfrom a risk minimizing perspective buying bonds with maturities close to theinvestment horizon seems optimal.329 Fong/Vasicek (1984) develop a measure

328 See Fabozzi (2004), p. 476.


of immunization risk called M-squared. The calculation of M-squared requiresonly information about the securities under consideration and the investmenthorizon.330 According to Bierwag/Fooladi/Roberts (1993) the minimum M-squared portfolio is a bullet portfolio under a specific convexity condition.331

In our Vasicek example the minimum-variance portfolio had the structure ofbullet portfolios as well. But in the HW2 model the structure of the minimum-variance portfolios didn’t resemble bullet portfolios. Agca (2002) has compa-rable findings in that only in some cases minimum M-squared portfolios werealso bullet portfolios.332

4.3.4 Summary and Conclusion

This part compared real-world bond portfolio selection methods with themean-variance model. In our numerical examples, most of the time, the resultsof the mean-variance optimization didn’t constitute portfolios that the prac-tice would have suggested in such an interest rate environment. The triggers(or factors) for constructing the portfolio in a specific way from a practi-tioner’s point of view hence seem to play a lesser role in the mean-varianceframework. Table 4.28 summarizes the results from this section.333

Table 4.28. Comparison with real-world portfolio selection methods.

Strategy Interest Rates Model Equal Table

Riding Unchanged Vasicek Yes 4.17

DurationFalling

Vasicek Yes 4.8HW2 Yes 4.14

RisingVasicek No 4.19HW2 No 4.21

Yield CurveStrategies

SteepeningVasicek Yes 4.6HW2 Yes 4.12

FlatteningVasicek No 4.22HW2 No 4.23

Immunization FallingVasicek Yes 4.24HW2 Yes 4.26

When compared to the riding strategy, mean-variance efficient portfoliosproduced nearly the same results as the proposed real-world portfolios. Butthe portfolios advocated by the other active strategies could not in every case

329 At least in the Vasicek (1977) model.330 See Fong/Vasicek (1984), p. 1543.331 M-squared is a convex function of duration, see Agca (2002).332 See Agca (2002), p. 74.333 Column 4 indicates whether the results of the mean-variance optimization coin-

cided with the practical investment advice.


be generated inside the mean-variance framework. An expectation of fallingand rising interest rates produced mean-variance efficient portfolios with aMacaulay duration that was greater than the investment horizon. The dura-tion strategy recommends different Macaulay durations for different interestrate scenarios. The recommended portfolios of the yield curve strategies (i.e.bullet or barbell portfolios) could also not be reproduced. A steepening or flat-tening term structure lead to mean-variance efficient portfolios that resembledbullet portfolios.

The immunization strategy on the other hand, generated portfolios thatdiffered only insignificantly from the mean-variance efficient portfolio, espe-cially in the Vasicek model. From a practical point of view, the durationimmunized portfolios perform also quite well when we consider more reason-able term structure movements. This is surely due to the fact, that the levelfactor in a principal component analysis of the term structure explains mostof the variability in interest rates.334

A suggestion for future research is the analysis of the performance of mean-variance efficient portfolios and active bond portfolio selection techniquesunder different scenarios. We suspect that mean-variance efficient portfolioswould turn out to be superior, because in this framework the uncertainty re-garding the outcome is already being accounted for. One already constructsportfolios in a way that a different outcome than was expected doesn’t havea devastating influence on the portfolio. Active bond portfolio strategies onthe other hand assume that a particular interest rate scenario is going tomaterialize.

Another important passive portfolio selection technique is index tracking.It calls for constructing a portfolio in such a way as to match a benchmarkportfolio as closely as possible.335 In practice the benchmark is often a well-known index.336 For a comprehensive introduction to bond index funds seeMossavar-Rahmani (1991) and for a critical discussion see Granito (1987). Insuch a framework the appropriate measure of risk is not the standard deviationof portfolio returns but the so-called active return or tracking error.337 Hence,the objective of the investor is to minimize the tracking error over the nextperiod by adjusting the portfolio weights relative to the benchmark weights.Since this period is generally quite small (could be a day, or month), we donot need a reinvestment assumption.

The general optimization problem is then338

334 See e.g. the seminal study by Litterman/Scheinkman (1991).335 Fabozzi (2004), p. 452.336 For more information about bond index construction see Brown (1994).337 See Fabozzi (2004), p. 415.338 See Martellini/Priaulet/Priaulet (2003), p. 218.


minw

var(RP −RB)

s.t. w′ · 1 = 1w ≥ 0

where RP is the return of the portfolio, RB is the return of the benchmark,w is the portfolio weights vector and 1 is an appropriate vector of ones. Thevariance can be written as

w′Cw

where C is the covariance matrix of bond returns and w is the active portfolio,i.e. w = wB − w. As becomes obvious from the above formulation, when weset w = wB , then the tracking error becomes zero.339 In reality, the existenceof transaction costs usually forbids perfect replication. In contrast to stock in-dices, bond indices contain a very large number of bonds and furthermore theindex changes quite often due to the exclusion of maturing bonds.340 Hence, apossible objective is to replicate a bond index consisting of 5.000 bonds with100 bonds as good as possible thereby introducing an additional constraint tothe above formulation of the tracking error minimization problem.

The critical ingredient of any tracking error model is the covariance matrixof returns. Martellini/Priaulet/Priaulet (2003) propose to use the sample co-variance matrix of bond returns341 or a risk factor-based covariance matrix.342

We suggest using a dynamic term structure model to generate the covariancematrix of bond prices, as has been shown already in this chapter.

A full derivation and application of such a model is outside the scope ofthis thesis. But a possible line of future research is to compare the trackingerror differences resulting from different covariance-generating methods.

339 This is called replication, see Martellini/Priaulet/Priaulet (2003), p. 213.340 See Elton et al. (2003), p. 682.341 See Martellini/Priaulet/Priaulet (2003), p. 217.342 See Martellini/Priaulet/Priaulet (2003), p. 222.

5

Dynamic Bond Portfolio Optimization inContinuous Time

5.1 Introduction

The single-period model for bond portfolio optimization is an extreme simplifi-cation of reality. It remains nevertheless quite popular in real-life applicationsbecause it demands no special knowledge beyond basis probability theory.343

To overcome the obvious limitations of the single-period model, continuous-time models of asset allocation were developed in the 1970s.344

In a continuous-time portfolio selection problem, the investor maximizeshis expected utility of terminal wealth (and maybe expected utility of con-sumption during a specific time interval) by selecting an optimal portfoliostrategy, i.e. he chooses the number of assets i to hold at each time t (andin every state of the world). Generally it is assumed that the investor canallocate his wealth to a (locally) riskless money market account and differentrisky assets. In our setting the risky assets are zero-coupon bonds of differentmaturities.

The solution of optimal portfolio selection and consumption strategies incontinuous time dates back to Merton (1969) and Merton (1971). He wasthe first to apply the stochastic control (continuous-time equivalent of thedynamic programming) approach345 to a continuous-time optimal consump-tion/investment problem.346 Merton assumed that the interest rate is con-stant. In order to apply his solution technique to bond portfolio selectionproblems, the constant interest rate assumption must be dropped.

The stochastic control approach computes the optimal solution to the port-folio selection problem by solving the Hamilton/Jacobi/Bellman (HJB) equa-

343 See Korn (1997), p. 1.344 The basic continuous-time setting is developed for example in Korn (1997),

Munk (2004a), pp. 33 ff. or Bjork (1998), pp. 198 ff.345 Another popular approach for solving the continuous-time asset allocation prob-

lem is the martingale approach introduced by Pliska (1986) and Cox/Huang(1989).

346 Munk (2004a), p. 36.

86 5 Dynamic Bond Portfolio Optimization in Continuous Time

tion347 in two steps: the first step entails computing the optimal strategy asa function of the unknown optimal expected utility and the second step con-sists of inserting this strategy into the HJB equation.348 In order to applythe stochastic control approach, we must assume the existence of a finite-dimensional Markov state process xt such that the indirect utility function349

J can be written as J(t,Wt, xt), where Wt is the time t wealth.350 In contrastto the martingale approach, one doesn’t have to assume a complete market.351

There exists a growing literature on bond portfolio selection in continuoustime. The following table summarizes the existing literature.352

Table 5.1. Existing literature on continuous-time bond portfolio selection.

Model Term Structure Models Utility functions Approach

Sørensen (1999) Vasicek (1977) u(W ) = W1−γ−11−γ

MA

Korn/Kraft (2002) Ho/Lee (1986), Vasicek(1977)

u(W ) = W γ SCA

Munk/Sørensen (2004) Heath/Jarrow/Morton(1992)

u(W ) = eβT W1−γ−11−γ

MA

Kraft (2004) Ho/Lee (1986), Vasicek(1977), Dothan (1978),Black/Karasinksi(1991), Cox/Ingersoll/Ross (1985)

u(W ) = W γ SCA

In this section we want to derive the solution to the bond portfolio selectionmodel for the Vasicek model as in Korn/Kraft (2002) and want to add tothe literature by deriving an explicit solution for the HW2 model using thestochastic control approach. In the last part of this chapter, we look at aninternational bond portfolio selection problem and derive an explicit solutionfor a two-country problem.

347 Will be introduced later in this chapter.348 See Korn (1997), p. 37.349 This function will be properly derived later.350 Munk (2004a), p. 40.351 See Korn (1997), p. 37.352 In column 2 the examined term structure model for modeling the bond markets

are given, column 3 gives the utility function of the investor and the last col-umn indicates the approach used (MA = martingale approach, SCA = stochasticcontrol approach).

5.2 Bond Portfolio Selection Problem in a HJM Framework 87

5.2 Bond Portfolio Selection Problem in a HJMFramework

The Heath/Jarrow/Morton (1992) framework introduced in Chapter 3.3 isthe most general term structure model. The problem of optimal consumptionand investment strategies within a general HJM-framework was addressedby Munk/Sørensen (2004) using the martingale approach.353 In order to usethe traditional stochastic control approach, we restrict our attention to a lessgeneral term structure model.

5.2.1 Dynamics of Prices and Wealth

We assume that the shifts in the investment opportunities are (only) drivenby a k-dimensional Markov diffusion process x governed by the SDE

dx(t) =α(x, t)dt + β(x, t)dz(t) (5.1)

where α(x, t) is the (k×1) drift vector, β(x, t) is the (k×d) matrix of volatil-ities354 and dz(t) is a (d × 1) vector of Brownian motions. This ensures thatwe can find a discrete set of variables that drives the shifts in the investmentopportunity set and satisfies the Markov property.

The investor can choose among n zero-coupon bonds of different matu-rities and a locally riskfree money market account. He can hence invest hisfunds in q = n + 1 different assets. First, we derive the joint dynamics of thezero-coupon bond prices. With the zero-coupon bond price dynamics fromEquation (3.5) and the arbitrage-free condition from Equation (3.13), we canwrite the dynamics of the n zero-coupon bond prices in matrix notation asfollows

dPt = diag(Pt) ((rt1n + σtλt)dt− σtdz(t)) (5.2)

where Pt designates the (n× 1) vector of zero-coupon bond prices at time t,diag(Pt) is a (n × n)-diagonal matrix of zero-coupon bond prices, rt is thevalue of the short rate at time t, λt is a (d×1) vector of market prices of risk,1n is a (n× 1) vector of ones and σt is a (n× d) matrix of volatilities.355

353 See Table 5.1.354 βi,j is the volatility of the i-th state variable with respect to the j-th Brownian

motion.355 The matrix σt is hence constructed as follows:

σt =

σ1(t, T1) . . . σd(t, T1)...

......

σ1(t, Tn) . . . σd(t, Tn)


The SDE for the money market account was already determined in equa-tion (3.3). The money market account (locally riskless asset) is hence governedby

dBt = Btrtdt (5.3)

The investor’s wealth at time t equals the sum of his portfolio holdings inthe q different assets

W (t) = N ′t

(Bt

Pt

)

Nt is the (q × 1) holdings vector at time t. P is the (n × 1) vector of zero-coupon bond prices at time t and B is the price of the money market account.Using Ito’s Lemma, we obtain the wealth dynamics

dW (t) = N ′t

(dBt

dPt

)+ dN ′

t

(Bt + dBt

Pt + dPt

)

The first term describes the change in the portfolio value due to changesin prices. The second term describes the change in portfolio value due tochanges in the portfolio holdings (rebalancing). We restrict our attentionto self-financing portfolios, hence the second term must be equal to zero.356

Hence, we obtain the following equation for the wealth dynamics

dW (t) = N ′t

(dBt

dPt

)

It is advantageous to replace the quantities vector Nt with a formulationcontaining portfolio weights. We introduce the vector of portfolio weights w357

w = (wB , w′)′

where the scalar wB denotes the portfolio weight of the money market accountand the (n× 1) vector w contains the portfolio weights of the n zero-couponbonds. There exists the following relationship between numbers of bonds andfractions of wealth

w′ =1

W (t)N ′

t

(Bt 00 diag(Pt)

)

We solve for N ′t and obtain

N ′t = w′W

(B−1

t 00 diag(Pt)−1

)

Inserting N ′t in dW (t) we obtain

356 The investor on the one hand does not receive exogenous income and on the otherhand doesn’t consume. The change in wealth during an infinitesimal time intervalis therefore caused only by changes in prices.

357 For notational convenience we ignore the time-dependence of w.


dW (t) = w′W (t)(

1Bt

00 diag(Pt)−1

)(dBt

dPt

)

= w′W (t)(

1Bt

00 diag(Pt)−1

)(Btrtdt

diag(Pt)(rt1n + σtλt)dt− σtdzt

)

= w′W (t)(

rtdt(rt1n + σtλt)dt− σtdzt

)

= (wB , w)(

rtdt(rt1n + σtλt)dt− σtdzt

)W (t)

= W (t)wBrtdt + W (t)w′((r11n + σtλt)dt− σtdzt)

Since portfolio weights must sum to one, we can eliminate wB using the fol-lowing equation

wB = 1− w′1n

We insert wB into dW (t), simplify and obtain the dynamics of the investor’swealth

dW (t) = W (t) ((rt + w′tσtλt)dt− w′tσtdzt) (5.4)

with W (0) = W0. This equation can be interpreted as a controlled SDE withthe control being the portfolio process w.358

5.2.2 The Hamilton/Jacobi/Bellman Equation

In this setting, the investor chooses a portfolio process to maximize his ex-pected utility of terminal wealth.

maxw

Et [u(W (T ))] (5.5)

We define the optimal value function359 J(W (t), xt, t)

J(W (t), xt, t) ≡ maxw

Et [u(W (T ))] (5.6)

Our derivation of the Hamilton/Jacobi/Bellman equation follows Ingersoll(1987) and Kamien/Schwartz (1981).360 With one period ∆t to the investmenthorizon, the investor solves the problem

J(W (T −∆t), x(T −∆t), T −∆t) = maxw

ET−∆t [J(W (T ), xT , T )] (5.7)

358 Korn/Kraft (2002), p. 1252.359 Also referred to as the indirect (or derived) utility function. Since we are operating

in a market with stochastic investment opportunities the function J depends –besides t and W – on the vector of state variables x.

360 See Ingersoll (1987), pp. 271–274 and Kamien/Schwartz (1981), pp. 246–247.


Continuing in a recursive fashion, we eventually obtain361

J(W (t), xt, t) = maxw

Et [J(Wt + ∆Wt, xt + ∆xt, t + ∆t)] (5.8)

We do a Taylor series expansion of J around (t,W, x1, . . . , xd) to get362

J(Wt + ∆Wt, x1 + ∆x1, . . . , xd + ∆xd, t + ∆t) =

J(t,Wt, x1, . . . , xd) + Jt∆t + JW ∆W +d∑

i=1

Jxi∆xi +d∑

i=1

JxiW ∆xi∆W

+12JWW ∆W 2 +

12

d∑

i=1

d∑

j=1

Jxixj∆xi∆xj (5.9)

This result can be written more readable in matrix notation363

J(Wt + ∆Wt, x + ∆x, t + ∆t) =J(t,Wt, xt) + Jt∆t + JW ∆W + J ′x∆x + J ′xW (∆x)(∆W )

+12JWW (∆W )2 +

12tr(Jxx′(∆x)(∆x′))

where

Jx =

Jx1

...Jxd

, JxW =

Jx1W

...JxdW

, Jxx′ =

Jx1x1 . . . Jx1xd

......

Jxdx1 . . . Jxdxd

We calculate the expectation

E[J(W + ∆W,x + ∆x, t + ∆t)] =J(W,x, t) + Jt∆t + JW E[∆W ] + JxE[∆x] + JxW E[∆x∆W ]

+12JWW E[∆W 2] +

12tr(Jxx′E[∆x∆x′])

and insert this expression into Equation (5.8). We obtain

0 = maxw

(Jt∆t + JW E[∆W ] + J ′xE[∆x] + J ′xW E[∆x∆W ]

+12JWW E[∆W 2] +

12tr(Jxx′E[∆x∆x′])

)(5.10)

Now we calculate the necessary expectations. With ∆x from (5.1) and ∆Wfrom (5.4) we obtain364

361 See Ingersoll (1987), p. 273.362 Apply the following multiplication rules for Brownian motions ∆t2 = 0, ∆t∆xi =

0 and ∆t∆W = 0, see Malliaris/Brock (1982), p. 87.363 tr(M) designated the trace of the matrix M .364 For a derivation see Appendix B.


E[∆x] = α∆t

E[∆W ] = (rt + w′σλ)W∆t

E[∆x∆x′] = ββ′∆t

E[∆x∆W ] = −βσ′wW∆t

E[∆W 2] = (w′σσ′w)W 2∆t

We insert these expressions into (5.10) and obtain

0 = maxw

(Jt∆t + JW (rt + wT σλ)W∆t + J ′xα∆t− J ′xW (βσ′w)W∆t

+12JWW (w′σσ′w)W 2∆t +

12tr(Jxx′ββ′)∆t

)

We divide by ∆t, let ∆t → 0 and eventually obtain the HJB equation365

0 = maxw

(Jt + JW (rt + wT σλ)W + J ′xm− J ′xW (sσ′w)W

+12JWW (w′σσ′w)W 2 +

12tr(Jxx′ss

′))

(5.11)

with boundary condition

J(W (T ), xT , T ) = maxw

ET [u(W (T ))]

= u(W (T )) (5.12)

5.2.3 Derivation of Optimum Portfolio Weights

For notational convenience Equation (5.11) can be written as follows

0 = maxw

(φ(w; W,x, t)) (5.13)

We get the n first-order conditions for an optimum by calculating the deriva-tive of φ w.r.t. w

φw = WσλJW −Wσs′JxW + σσ′w∗W 2JWW = 0 (5.14)

where 0 is a (n× 1) vector of zeros. Solving for w∗ we obtain

w∗ =(− JW

WJWW

)(σσ′)−1σλ + (σσ′)−1σβ′

JxW

WJWW

We define Σ ≡ σσ′ and Σ ≡ σβ′ and can the solution in the following shorterform

w∗ =(− JW

WJWW

)Σ−1σλ + Σ−1Σ

JxW

WJWW(5.15)

365 See Ingersoll (1987), p. 282.


The necessary conditions are366

φww < 0det(φww) > 0

Sinceφww = W 2JWW Σ

and Σ is positive-definite, we have a single necessary condition for an interiormaximum, namely JWW < 0.367

Number of Zero-Coupon Bonds to Include in Bond Portfolio

At the beginning of this section, we simply stated that the investor can investin n zero-coupon bonds of different maturity and in a money market account.We didn’t elaborate on how many different bonds the investor really needs toconsider. That depends on the interest rate model under consideration and isclosely related to the question of duplicable cash flows. The optimum portfolioweights from Equation (5.15) are

w∗ =(− JW

WJWW

)Σ−1σλ + Σ−1Σ

JxW

WJWW

This equation has a solution if the inverse of Σ exists. The matrix Σ can bewritten as follows

Σ =

∑di=1 σi(t, T1)2

∑di=1 σi(t, T1)σi(t, T2) ...

∑di=1 σi(t, T1)σi(t, Tn)

......

...∑di=1 σi(t, Tn)σi(t, T1)

∑di=1 σi(t, Tn)σi(t, T2) ...

∑di=1 σi(t, Tn)2

This is the symmetric covariance matrix of instantaneous zero-coupon bondreturns

Σ =

var(

dP (t,T1)P (t,T1)

)cov

(dP (t,T1)P (t,T1)

, dP (t,T2)P (t,T2)

)... cov

(dP (t,T1)P (t,T1)

, dP (t,Tn)P (t,Tn)

)

......

cov(

dP (t,Tn)P (t,Tn) , dP (t,T1)

P (t,T1)

)cov

(dP (t,Tn)P (t,Tn) , dP (t,T2)

P (t,T2)

)... var

(dP (t,Tn)P (t,Tn)

)

The inverse of a quadratic matrix exists if the matrix has a determinant thatis not zero.368 The determinant of a matrix is zero if369

366 See Merton (1992), p. 129.367 See Merton (1992), p. 129.368 See Bronstein et al. (1999), p. 255.369 See Bronstein et al. (1999), p. 259.


1. One row consists only of zeros.2. Two rows are identical.3. One row is a linear combination of other rows.

We now consider each of the different cases. Case 1 can only occur, if weinclude a (locally) riskless asset in the zero-coupon bond set. A (locally) risk-less asset is a zero-coupon bond with instantaneous maturity. This bond is(economically) equivalent to the money market account. The money marketaccount is considered separately and is therefore not included in the covariancematrix of instantaneous zero-coupon bond returns. Hence case 1 is precluded.

The occurrence of two identical rows in the covariance matrix happensto coincide with the situation that two assets are identical. Two zero-couponbonds are identical if they have the same maturity. This is also precluded.

Last but not least we consider case 3. Now the concept of stochastic lineardependence is useful. It is well known that there can be only so many stochas-tic linear independent assets as there are risk factors (Brownian motions).In our framework, there are d Brownian motions that drive the underlyinguncertainty. Therefore, there can be a maximum of d stochastic linear inde-pendent assets. Hence, the covariance matrix is invertible if the zero-couponbonds included in the investment opportunity set consist of a maximum of dzero-coupon bonds with different maturity.

In reality, more than d zero-coupon bonds trade in the market370 hencewe can choose a subset of these traded bonds. Economically this means, thatd different zero-coupon bonds and a position in the money market accountcan dynamically replicate all interest rate dependent securities (and hence allother zero-coupon bonds).

Economic Interpretation of Optimum Portfolio Weights

The optimum portfolio weights in Equation (5.15) have been subject to far-ranging analysis from different authors. We first summarize these findings andthen look for differences between the equity market interpretations and thebond market.

The optimum portfolio weights consist of a mean-variance-efficient port-folio and k hedge portfolios.371 The fact that

(− JW

WJWW

)Σ−1σλ

is mean-variance efficient and Σ−1σλ is the Tobin-Fund has been shown byIngersoll (1987).372 Since J depends among other things on the time from t

370 Even in a multi-factor term structure model d is usually quite small in accordancewith empirical research from e.g. Litterman/Scheinkman (1991). They concludedthat d should be no greater than 3, i.e. three factors account for nearly 100 % ofthe variability in interest rates.

371 See Nietert (1996), p. 54.372 For a derivation see Ingersoll (1987), p. 283.


to T , a simultaneous optimization takes place and no sequential (“myopic”)decision making.373

In this general model, the investment opportunity set is stochastic, i.e.changes in the state variables change the nature of the available assets. Inger-soll (1987) finds that the hedge portfolios374 provides the best possible hedgeagainst change in the state variables since it has the maximum absolute cor-relation with it.375 We introduce the following notation. Let Σi denote thei-th column of Σ, then we can write (5.15) as follows376

w∗ =(− JW

WJWW

)Σ−1σλ +

k∑

i=1

Σ−1ΣiJxiW

WJWW

where

Σi =(

var(

dP (t, T1)P (t, T1)

, dxi

), . . . , var

(dP (t, Tn)P (t, Tn)

, dxi

))′

A hedge portfolio for a state variable can hence only be constructed if thestate variable is correlated with at least one asset.377 For a thorough analysisof the hedge portfolios in a two asset and one state variable case see Ni-etert (1996), pp. 55–59. We have a further look at the economic interpretationof the optimum portfolios in a later section.

5.2.4 The Value Function for CRRA Utility Functions

The optimum portfolio weights in Equation (5.15) still depend on the unknownfunction J . Finding a solution for J requires the insertion of the optimumweights in the HJB, we obtain

Jt + JW rW + J ′xα +12tr(Jxx′ββ′)− 1

2J2

W

JWW(σλ)′(σσ′)−1(σλ)

+JW

JWWJ ′xW βσ′(σσ′)−1σλ− 1

21

JWWJ ′xW βσ′(σσ′)−1σβ′JxW = 0 (5.16)

Simplification yields378

Jt + JW rW + J ′xα +12tr(Jxx′ββ′)− 1

2J2

W

JWW(σλ)′(σσ′)−1(σλ)

+JW

JWWJ ′xW βλ− 1

21

JWWJ ′xW ββ′JxW = 0 (5.17)

373 See Nietert (1996), p. 20.374 Σ−1Σ JxW

WJW W375 See Ingersoll (1987), p. 282.376 See Nietert (1996), p. 54.377 See Nietert (1996), p. 54.378 σ(σσ′)−1σ = Idd, where Idd is the (d× d) identity matrix.


We multiply with JWW and obtain

JtJWW + JWW JW rW + JWW J ′xα +12JWW tr(Jxx′ss

′)

− 12J2

W (σλ)′(σσ′)−1(σλ) + JW J ′xW βλ− 12J ′xW ββ′JxW = 0 (5.18)

Unfortunately the HJB is usually highly non-linear and hence explicit solu-tions are only available in some cases. In accordance with the existing lit-erature on continuous-time bond portfolio selection379 we assume that thepreferences of the investors can be represented by a utility function with con-stant relative risk aversion (CRRA) of the form

u(W (T )) = W γ (5.19)

with 0 < γ < 1.380 We now conjecture that the unknown value function canbe separated as follows381

J(W (t), xt, t) = f(x, t)u(W (T )) = f(x, t)W γ (5.20)

We insert the partial derivatives382 of J(W (t), xt, t) into (5.18) and after re-ducing W 2γ−2 and γ, we obtain a PDE for the unknown function f(x, t)

(γ − 1)ftf + γ(γ − 1)f2r + (γ − 1)ff ′xα +12(γ − 1)ftr(fxx′ββ′)

− 12γf2(σλ)′(σσ′)−1(σλ) + γff ′xβλ− 1

2γf ′xββ′fx = 0 (5.21)

with boundary condition f(xT , T ) = 1. With (5.21) we are in a position tofind a solution for the unknown value function J(W (t), xt, t).

With J(W (t), xt, t) from Equation (5.20), we obtain the following expres-sion for the optimum portfolio weights in the CRRA case

w∗ =1

1− γ

(Σ−1σλ−Σ−1Σ

fx

f

)(5.22)

379 See Table 5.1.380 Korn/Kraft (2002), p. 1252.381 Analogous to Korn (1997), p. 52 or Korn/Kraft (2002), p. 1254.382 The partial derivatives are

Jt = ftWγ

Jx = fxW γ

JW = γW γ−1f

JWW = γ(γ − 1)W γ−2f

Jxx = fxxW γ

JxW = γW γ−1fx


In the next section, we want to derive two special cases. First we want toexamine dynamic bond portfolio selection in a Vasicek model and then derivethe optimum portfolio weights in a HW2 model.

5.3 Special Cases

5.3.1 One-Factor Vasicek (1977) Model

The dynamic bond portfolio selection problem in a Vasicek (1977) term struc-ture model has already been solved by Korn/Kraft (2002) using the stochasticcontrol approach. We derive their result as a special case of Equation (5.22).

Derivation the Optimum Portfolio Weights

In the Vasicek model the only state variable is the short rate of interest,i.e. x(t) = r(t) and k = 1. Furthermore, the short rate is influenced by oneBrownian motion only (d = 1). As we have shown in Section 5.2.3 the numberof zero-coupon bonds that we have to consider depends on the number ofBrownian motions. In this case, the optimum portfolio consists of holdingsin the riskless money market account and in one risky zero-coupon bond ofmaturity T1 (n = 1).

With k = d = n = 1, Equation (5.22) reduces to

w∗t =1

1− γ

(λ

σ(t, T1)− β(t)

σ(t, T1)fr

f

)(5.23)

We specify the function f introduced in Equation (5.20) as follows383

f(r, t) = g(t) exp(v(t)r) (5.24)

thenfr

f= v(t)

and the portfolio weights in Equation (5.22) become

w∗ =1

1− γ

(λ

σ(t, T1)− β(t)

σ(t, T1)v(t)

)(5.25)

In the Vasicek model the volatility of the short rate is a constant, i.e.β(t) = σr.384 Furthermore, the volatility of a bond maturing at time T1 isσ(t, T1) = σrB(t, T1) = σr

(1−e−κ(T1−t)

κ

).385

383 See Korn/Kraft (2002), p. 1254.384 See Equation (3.16).385 See Korn/Kraft (2002), p. 1252.

5.3 Special Cases 97

We insert (5.24) in (5.21), set the drift of the short rate equal to386

α = κ(θ − r) and obtain

e2rv(t)r(γ − 1)g(t)2 (γ − κv(t) + v′(t))− 12e2rv(t)g(t)

× (g(t)

(γλ2 + σ2

rv(t)2 − 2((γ − 1)θκ + σrγλ)v(t))

−2(γ − 1)g′(t)) = 0 (5.26)

subject to the boundary conditions v(T ) = 0 and g(T ) = 1.387 Since

r(γ − 1)g(t)2 (γ − κv(t) + v′(t)) = 0

must hold for all values of r, the term (γ − κv(t) + v′(t)) must be equal tozero. This is a second order ODE for the unknown function v(t) subject tov(T ) = 0. The solution is388

v(t) =

(1− e−κ(T−t)

)γ

κ(5.27)

It is interesting to note that

v(t) = γB(t, T )

With v(t) from (5.27), Equation (5.26) becomes an ODE for the unknownfunction g(t). The function g(t) is not needed for the optimum portfolio strat-egy, therefore we give its solution only in the Appendix.389 Hence, v(t) from(5.27) and g(t) from (C.1) solve (5.26). The optimal portfolio strategy canthen be written as390

w∗t =1

1− γ

(λκ

σr(1− exp(−κ(T1 − t)))− γ

1− exp(−κ(T − t))1− exp(−κ(T1 − t))

)

or

w∗t =1

1− γ︸︷︷︸− JW

W JW W

(λκ

σr(1− e−κ(T1−t))

)

− γ

1− γ

1− e−κ(T−t)

κ︸︷︷︸JrW

W JW W

(κ

1− e−κ(T1−t)

)(5.28)

386 See Equation (3.16).387 Since f(r, T ) = 1.388 See Korn/Kraft (2002), p. 1255.389 See Appendix C.1, Equation (C.1).390 See Korn/Kraft (2002), p. 1255.


Interpretation of Optimum Portfolio Weights

As has been discussed in the last section, the optimum portfolio strategy inEquation (5.28) consists of a mean-variance efficient portfolio and one hedgeportfolio. Furthermore, since all parameters are constant and the short ratehas no influence on the portfolio weights, it is a deterministic function oftime. We first examine the mean-variance efficient portfolio. As can be seenfrom Equation (5.28) it is independent of the investment horizon and can beregarded as a product of a volume and a structural component.

11− γ︸︷︷︸

volume component

(λκ

σr(1− e−κ(T1−t))

)

︸︷︷︸structural component

The volume component depends only on the risk aversion of the investor.It is positive391 and increasing in γ. The structural component reflects thetrade-off between expected risk premium to risk contribution.392

Next, we analyze the hedge portfolio. We have stated in Section 5.2.3 thatthe hedge portfolio has the maximum absolute correlation possible. In a onefactor model this is no defining characteristic since the hedge portfolio canconsist only of one zero-coupon bond and every zero-coupon bond has thesame instantaneous correlation with the state variable (short rate), namely−1.393 The hedge portfolio consists of a volume term and a structural com-ponent as well

γ

γ − 11− e−κ(T−t)

κ︸︷︷︸volume component

(1

1− e−κ(T1−t)

)

︸︷︷︸structural component

The volume component depends on the risk aversion of the investor, the in-vestment horizon and on some distributional parameters of the state variable(the short rate). It does not depend on the level of volatility σr and the marketprice of interest rate risk. As the investment horizon approaches, the hedgeportfolio position in the zero-coupon bond approaches zero since

limt→T

γ

γ − 11− e−κ(T−t)

κ= 0

This is economically plausible, since changes in the investment opportunityset have a smaller effect on the portfolio value as the investment horizonapproaches.

391 Since 0 < γ < 1.392 See Nietert (1996), p. 20.393 With dr from (3.16) and the dP from (3.5) one derives for the Vasicek model a

correlation coefficient % of −1. Economically speaking bond prices move inverselyto interest rates.


For the portfolio as a whole the following results can be derived. Themarket prices of interest rate risk λ determine the relative attractiveness of thezero-coupon bond relative to the money market account, it is straightforwardto show that ∂w∗t

∂λ > 0. The short rate volatility σr which determines thevolatility level for all other interest rates has a plausible effect too. If thevolatility declines (rises) than the optimum portfolio weight of the zero-couponbond rises (declines), i.e. ∂w∗

∂σ < 0 if λ > 0. The investor has no position inthe zero-coupon bond at time t if

1γ

λ

σr=

1− e−κ(T−t)

κ

Since the left hand side is constant and the right hand side is a function of t,such an occurrence seems to be quite rare.

Numerical Example

The practical application of the model is problematic since it is extremelysensitive to (unobservable) input parameters, i.e. estimation errors have quitea significant effect on the portfolio weights. To illustrate the effect, supposethat γ = 0.5, σ = 0.02, κ = 0.2, T − t = 5 and T1 − t = 10, i.e. the investmenthorizon is five years and the bond under consideration is a 10 year zero-couponbond. The following table gives the portfolio weight of the risky zero-couponbond at time t for different values of λ.

Table 5.2. Vasicek model: Zero-coupon bond weight for different market prices ofinterest rate risk.

λ 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

w∗ -0.73 -0.27 0.19 0.66 1.12 1.58 2.04 2.51 2.97 3.43 3.9

We assume that λ = 0.04 and we are now interested in the behavior of theoptimum portfolio weight as a function of time, in other words, how does thezero-coupon bond weight change as the investment horizon approaches? Wefind that the weight of the risky zero-coupon bond increases as we approachthe investment horizon.394 The portfolio weight as a function of time is shownin Figure 5.1.395

394 This behavior (i.e. rising risky bond weight as the investment horizon approaches)is not true in general, but only if eT1κγσ + eTκ(κλ − γσ) > 0. To arrive at thissolution calculate the partial derivative of w∗ w.r.t t and let this result be greateror equal to zero and simplify.

395 Furthermore for different values of λ this graph simply moves in parallel, this isobvious since the partial derivative of the optimum portfolio weights w.r.t. λ isconstant.


1 2 3 4 5t

0.2

0.4

0.6

0.8

1

1.2

w

Fig. 5.1. Vasicek Bond Portfolio Selection: Zero-Coupon Bond weight w as a func-tion of time

This suggested investment behavior is contrary to popular investment ad-vice. Typical investment advice is to decrease risky holdings as the investmenthorizon approaches.

Comparison with Classical Active Bond Portfolio Strategies

In the last chapter on static bond portfolio selection, we compared the result-ing portfolios to classical yield curve strategies. We refrain from comparingthe dynamic models for several reasons. First, in the dynamic models, the un-derlying bond market uncertainty determines to a large extent the portfoliostructure. Given a one-factor model, we need a position in the money marketaccount and a position in a zero-coupon bond with a maturity greater thanthe investment horizon. Hence, no ladder portfolios are possible. Second, theportfolio weights in the dynamic models can become negative as can be seenfrom Table 5.2. The real-world methods we introduced in the last chapter as-sumed that all portfolio weights are positive. But if we restrict short sales inthe dynamic models, then the problem becomes far more difficult to solve.396

5.3.2 Two-Factor Hull/White (1994) Model

In this section we want to examine bond portfolio selection in a HW2 model.Munk/Sørensen (2004) solve a dynamic bond portfolio selection problem inthe HJM framework using the martingale approach. We derive an explicitsolution for the HW2 model using the stochastic control approach and henceadd to the existing literature on continuous-time bond portfolio optimization.

Derivation of Optimum Portfolio Weights

The HW2 model has two state variables: the short rate of interest r(t)and the mean reversion level ε(t), hence the vector of state variables is

396 Constrained optimization problems in continuous-time have been studied by Cvi-tanic/Karatzas (1992), Xu/Shreve (1992a) and Xu/Shreve (1992b).


x(t)′ = (r(t), ε(t)). The underlying uncertainty is driven by two uncorrelatedBrownian motions z1(t) and z2(t). Hence – as has been shown in Section(5.2.3) – it is necessary to hold two zero-coupon bonds of maturities T1 andT2 and the money market account, i.e. k = d = n = 2. We can specify thequantities in the general portfolio weight Equation (5.22) as follows

σ =(

σ1(t, T1) σ2(t, T1)σ1(t, T2) σ2(t, T2)

)

λ =(

λ1

λ2

)

β =(

σr 0%σε

√1− %2σε

)

fx =(

fr

fε

)

A closed-form solution for the optimal portfolio weights can be found if weassume the following functional form for f(x, t). The function f(x, t) separatesthe indirect utility function J and has been introduced in Equation (5.20)

f(r, ε, t) = g(t)ep(t)r(t)+q(t)ε(t)

where p(t) and q(t) are deterministic functions of time, hence

fx

f=

1f

(fr

fε

)=

(p(t)q(t)

)

The zero-coupon bond volatilities σ1(t, T ) and σ2(t, T ) are397

σ1(t, T ) = σrB1(t, T ) + %σεB2(t, T )

σ2(t, T ) =√

1− %2σεB2(t, T )

Inserting the assumption for f in Equation (5.21) leads to

− erp(t)+εq(t)g(t)(γλ2

1 + γλ22 + σ2

rp(t)2 + σ2ε q(t)2

)

2(γ − 1)+ erp(t)+εq(t)g(t)

×γ

(%λ1 +

√1− %2λ2

)σεq(t) + p(t) ((γ − 1)θ + γλ1σr − %σrσεq(t))

γ − 1

+ erp(t)+εq(t)g′(t) + erp(t)+εq(t)rg(t) (γ − κrp(t) + p′(t))+

+ erp(t)+εq(t)εg(t) (p(t)− κεq(t) + q′(t)) = 0 (5.29)

It can be seen from (5.29) that in order for a solution to exist, the functionsp(t) and q(t) must meet the following ODEs

397 See Equations (3.7),(3.36) and (3.37).


(γ − κrp(t) + p′(t)) = 0 (5.30)

and−e(t−T )κrγ + γ + κr (q′(t)− κεq(t)) = 0 (5.31)

We first solve the ODE for the unknown function p. Its solution is

p(t) =

(1− e−(T−t)κr

)γ

κr(5.32)

We now insert the solution for p(t) into the second ODE for q(t) and obtain398

q(t) =γ

((1− e(t−T )κε

)κr +

(−1 + e(t−T )κr)κε

)

κr (κr − κε) κε(5.33)

We notice from (5.32), (5.32), (3.45) and (3.46) that the following rela-tionships hold

p(t) = γB1(t, T ) (5.34)q(t) = γB2(t, T ) (5.35)

We insert p(t) and q(t) into (5.29) and note that it becomes an ODE for theunknown function g(t). The solution to the function g(t) is once again notneeded for the computation of the optimum portfolio weights, hence we giveits solution only in the Appendix.399 p(t) from (5.33), q(t) from (5.33) andg(t) from (C.2) hence solve Equation (5.29). Now, we are able to give thesolution to the optimum zero-coupon bond portfolio weights. We insert p(t)from (5.34) and q(t) from (5.35) into Equation (5.22). We obtain

w∗ =1

1− γ

(Σ−1σλ

)+

γ

γ − 1

(Σ−1Σ

(B1(t, T )B2(t, T )

))

or

w∗ =1

1− γ

(Σ−1σλ

)−2∑

i=1

γ

1− γBi(t, T )Σ−1Σi (5.36)

where Σi is again the i-th column of the matrix Σ ≡ σβ′. The optimumportfolio weights are again an affine combination of three portfolios – themean-variance efficient portfolio and two hedge portfolios.

Interpretation of Optimum Portfolio Weights

Since the structure of the mean-variance efficient portfolio is the same as inthe Vasicek case, we concentrate on analyzing the hedge portfolios. The hedge

398 We can find a solution for g(t) too, but since this function is not necessary forthe calculation of the optimum portfolio weights we refrain from giving it here.

399 See Appendix C.2, Equation (C.2).


portfolio against the factor i (f1 = r and f2 = ε) can be written in detail asfollows

− γ

1− γBi(t, T )

var(

dP2P2

)cov

(dP1P1

,dfi

)−cov

(dP1P1

,dP2P2

)cov

(dP2P2

,dfi

)

var(

dP1P1

)var

(dP2P2

)−cov

(dP1P1

,dP2P2

)2

var(

dP1P1

)cov

(dP2P2

,dfi

)−cov

(dP1P1

,dP2P2

)cov

(dP1P1

,dfi

)

var(

dP1P1

)var

(dP2P2

)−cov

(dP1P1

,dP2P2

)2

(5.37)

with the approaching of the investment horizon T , the hedging volume400

becomes smaller since ∂Bi(t,T )∂t < 0 and it vanishes when t = T . It is interesting

to note from Equation (5.37) that the correlation between the factors r andε seems to play no explicit role. Nevertheless % has seemingly an influenceon the various covariances. But if we calculate the different covariances andinsert them into (5.37), the expression reduces to

− γ

1− γB1(t, T )

(B2(t,T2)

B1(t,T1)B2(t,T2)−B1(t,T2)B2(t,T1)B2(t,T1)

B1(t,T2)B2(t,T1)−B1(t,T1)B2(t,T2)

)(5.38)

for the first factor (r) and to

− 11− γ

B2(t, T )

(B1(t,T2)

B1(t,T2)B2(t,T1)−B1(t,T1)B2(t,T2)B1(t,T1)

B1(t,T1)B2(t,T2)−B1(t,T2)B2(t,T1)

)(5.39)

for the second factor (ε), with B1(t, T ) and B2(t, T ) defined in equations(3.45) and (3.46) respectively. The hedge portfolios therefore don’t dependon the correlation coefficient between the two factors. Mathematically thisfollows since the covariances are all linear in % and therefore we can reduce% accordingly. Economically this is due to the fact that the term structuremodel is affine linear and that all zero-coupon bonds are influenced by bothfactors, i.e. there is no idiosyncratic risk.

The hedge portfolio volumes are always non-negative since B1(t, T ) ≥ 0and B2(t, T ) ≥ 0.401 The first inequality is easy to show, the second needsmore attention. To see this, note that B2(t, T ) can only be negative if (i) thenominator is positive and the denominator is negative or (ii) the nominatoris negative and the denominator is positive. The sign of the denominatordepends only on the relationship between κr and κε, if κr > κε then thedenominator is positive and vice versa. The sign of the nominator depends ona more complicated expression. If

1− e−(T−t)κε

1− e−(T−t)κr>

κε

κr

then the nominator is positive and vice versa. It can then be shown that theabove mentioned constellations for a negative function B2(t, T ) cannot occurfor permissible values of the parameters.

400 − γ1−γ

Bi(t, T )401 And 0 < γ < 1.


Another interesting finding is that the signs of the portfolio weights foreach hedge portfolio are different, i.e. if the hedge portfolio for factor fi con-tains a long (short) position in the T1-zero-coupon bond, then it also containsa short (long) position in the T2-zero-coupon bonds. To see this, note that thefollowing relationship holds for denominators in the asset weights formula

B1(t, T1)B2(t, T2)−B1(t, T2)B2(t, T1) =− [B1(t, T2)B2(t, T1)−B1(t, T1)B2(t, T2)]

Comparing the zero-coupon bond weights from Equations (5.38) and(5.39), we also note that if there is a long (short) position in the τ -zero-coupon bond in the r-hedge portfolio, there must be a short (long) positionin this bond in the ε-hedge portfolio.402

Numerical Example

We want to conclude with a numerical example. We assume the same param-eters for the HW2 model as in the corresponding section in Chapter 4.2.5 onMarkowitz Portfolio Selection, i.e. r(0) = 0.025, % = 0.6, λ1 = 1.2395, λ2 = 0,σr = 0.0073, σε = 0.0219, κr = 0.2591, κε = 0.8274 and θ = 0.0053.

For this parametrization with an investment horizon of T = 5, two riskyzero-coupon bonds with maturity T1 = 10 and T2 = 30 and a risk aversionparameter γ = 0.5, we obtain the following portfolio weights at time 0:

w∗ =(

2593.44−2312.79

)(5.40)

This result can be disaggregated as follows:

Table 5.3. HW2 model: Disaggregation of optimum portfolio.

Asset Mean-Variance Hedge Portfolio 1 Hedge Portfolio 2 Sum

10-year zero 2,596.91 -21.43 17.97 2,593.44

30-year zero -2,315.27 19.11 -16.63 -2,312.79

MMA — — — -279.65

It is highly unlikely that such a portfolio would or could be implementedin practice since it requires huge long and short positions. In this respect, theresult is similar to the unrestricted static model. The reason is the extremedependence on the market prices of interest rate risk, λ1 and λ2. Even smallchanges in λ1 or λ2 have a pronounced effect on the portfolio weights.403

402 Compare again the denominators and use the fact that B1(t, T ) and B2(t, T ) arenon-negative.

403 The table gives the portfolio vector (w1, w2, w0) where w1 (w2) is the weight ofthe 10- year zero-coupon bond (30-year zero-coupon bond) and w0 is the weightof the money market account.

5.4 International Bond Investing 105

Table 5.4. HW2 model: Effect of market prices of interest rate risk on portfolioweights.

λ2 / λ1 0 0.05 0.1 0.15 0.2

0

−3.472.481.98

−118.15105.9613.2

−232.84209.4324.41

−347.53312.9135.63

−462.22416.3846.84

0.05

101.29−90.91−9.38

−13.412.561.84

−128.09116.0413.05

−242.78219.5124.27

−357.46322.9935.48

0.1

206.05−184.31−20.74

91.36−80.84−9.52

−23.3322.641.69

−138.02126.1212.9

−252.71229.5924.12

0.15

310.8−277.71−32.1

196.11−174.23−20.88

81.43−70.76−9.67

−33.2632.721.54

−147.95136.1912.76

0.2

415.56−371.1−43.46

300.87−267.63−32.25

186.18−164.15−21.03

71.49−60.68−9.82

−43.242.81.4

This table summarizes the portfolio weight sensitivities to changes in mar-ket prices of interest rate risk. In this numerical example, even small changesin the market prices have a huge impact on the portfolio weights. But weobserve more structure. For a given λ2, a higher value of λ1 leads to lowerweights for the shorter bond and higher weights for the longer bond. For agiven λ1, higher values of λ2 lead to higher weights for the shorter bond andlower weights for the longer bond. No values of λ1 and λ2 produce sensibleportfolio weights (0 ≤ wi ≤ 1) in this example.

The obvious solution of introducing short sale constraints into the opti-mization problem unfortunately makes the problem much more difficult sosolve. Constrained optimization problems in continuous-time have been stud-ied by Cvitanic/Karatzas (1992), Xu/Shreve (1992a) and Xu/Shreve (1992b).We will not solve this constrained optimization problem here.

5.4 International Bond Investing

5.4.1 Introduction

The general dynamic bond portfolio selection problem presented in the lastsections can be easily adapted to other settings. A theoretically interestingand practically important extension is to consider foreign investments, i.e.the inclusion of foreign currency bonds.404 Frequently, the investors don’thave regional investment restrictions and hence can invest not only in EUR-denominated Government bonds but also in Government bonds denominated

404 The general foreign currency framework is due to Amin/Jarrow (1991).


in a foreign currency, e.g. US Treasury Bonds or Japanese Government Bonds.Modern portfolio theory propagates the inclusion of non-perfectly correlatedassets in the investment universe because this generates diversification ben-efits. Foreign currency Government bonds are then a natural choice for theinvestor.

One major difficulty of portfolio optimization problems in an internationalsetting is modeling all asset prices in an arbitrage-free manner. In this sectionwe base the setup of the foreign exchange market on Lipton (2001). He usedcomparable frameworks to study the valuation of foreign exchange deriva-tives. We use his modeling approach to study an international bond portfolioselection model.405

5.4.2 Model Setup

There exist many possibilities of modeling international bond markets. Onecan decide on the number of countries one wishes to model, the term struc-ture of interest rate model in each country (one-factor, multi-factor) and thestochastic process for the foreign exchange rate. In this section we restrictour attention to a simple two-country model where the term structure in eachcountry is governed by the Vasicek term structure model.

• The investor can invest his funds in different assets in two countries406

– Country 1 (Germany, currency EUR) and– Country 2 (USA, currency USD)

• In every country the term structure of interest rates is determined bythe Vasicek (1977) term structure model with constant market prices ofinterest rate risk as introduced in Chapter 3.4. The short rate dynamicsare– Country 1

dr1(t) = κ1(θ1 − r1)dt + σr,1dz1 (5.41)

– Country 2dr2(t) = κ2(θ2 − r2)dt + σr,2dz2 (5.42)

We assume that dz1 and dz2 are uncorrelated.• The EUR/USD exchange rate S(t) is governed by a two-dimensional Ito

process. The exchange rate follows

dS(t)S(t)

= a(.)dt + b1(.)dz1 + b2(.)dz2 (5.43)

where a is the drift and b1 and b2 are the volatilities of the exchange rate.

405 To our knowledge this international bond portfolio selection model presented here,constitutes an addition to the existing literature.

406 For ease of exposition these countries are taken to be Germany and the UnitedStates of America.


First, we examine Germany (country 1). In Germany, a money market accounttrades. The dynamics of the money market account is given by the followingSDE

dM1(t)M1(t)

= r1(t)dt

where r1(t) is the (nominal) short rate of interest. Also zero-coupon bonds ofdifferent maturities trade with SDEs

dP1(t, T )P1(t, T )

= (r1(t) + σ1,1(t, T )λ1(t))dt− σ1,1(t, T )dz1

where λ1(t) is the z1-market price of risk in country 1 and σ1,1(t, T ) is thevolatility of the zero-coupon bond prices. Next, we consider the USA (country2). In the USA, there is also a market for a money market account. It followsthe SDE

dM2(t)M2(t)

= r2(t)dt

where r2(t) is the short rate of interest. There also trade zero-coupon bondsof different maturities. The SDEs for the bonds are

dP2(t, T )P2(t, T )

= (r2(t) + σ2,2(t, T )λ∗2(t))dt− σ2,2(t, T )dz2

where λ∗2(t) is the z2-market price of interest rate risk in USA and σ2,2(t, T )is the volatility of the zero-coupon bond prices.

We will see that since the currencies are freely convertible, the drift andthe volatilities of the exchange rate can’t be set independently of the bondmarkets. We now construct an arbitrage free international bond market, i.e.we find that drift a of the exchange rate that guarantees freedom of arbitrage.Our derivation follows Lipton (2001).407

First, we convert USD-prices to EUR

M∗(t) = M2(t)S(t)P ∗(t, T ) = P2(t, T )S(t)

The dynamics of both converted prices can be obtained by a straightforwardapplication of Ito’s lemma. The MMA dynamics are

dM∗(t)M∗(t)

= (r2 + a)dt + b1dz1 + b2dz2 (5.44)

The dynamics of the USD zero-coupon bond in EUR are obtained in a similarway

407 See Lipton (2001), pp. 223–227.


dP ∗(t, T )P ∗(t, T )= (r2(t) + a + σ2,2(t, T )(λ∗2(t)− b2))dt + b1dz1 + (b2 − σ2,2(t, T ))dz2

= (r2(t) + a + σ2,2(t, T )λ2(t))dt + b1dz1 + (b2 − σ2,2(t, T ))dz2 (5.45)

where we have defined λ2(t) := λ∗2(t)−b2. We now construct a riskless portfoliofrom holdings in P ∗, M∗ and P1. The instantaneous return of this risklessportfolio must be equal to the short rate in Germany (country 1), i.e. r1. Wesolve for a and obtain

a = r1 − r2 − λ1b1 − λ2b2 (5.46)

We can now insert a in the equations for S, P ∗ and M∗ but we can gainfurther insights if we examine 1/S first. The dynamics of S∗ = 1/S can againbe obtained in a straightforward manner by Ito’s lemma

dS∗ = a∗dt + b∗1dz1 + b∗2dz2

where

a∗ = −a + b21 + b2

2

b∗1 = −b1

b∗2 = −b2

Arbitrage-free considerations408 lead to the following functional form for a∗

a∗ = r2 − r1 − (λ1 − b∗1)b∗1 − λ∗2b

∗2

= r2 − r1 − (λ1 + b1)b∗1 − λ∗2b∗2

= r2 − r2 − λ∗1b∗1 − λ∗2b

∗2

where we defined λ∗1 := λ1 + b1. We now obtain the following relationships

b1 = λ∗1 − λ1 (5.47)b2 = λ∗2 − λ2 (5.48)

The foreign exchange volatilities are determined by the excess rates of returnon discount bonds and cannot be chosen independently of the bond marketsunder consideration.409 Equation (5.43) can hence be written as

dS(t)S(t)

= (r1 − r2 − λ1(λ∗1 − λ)− λ2(λ∗2 − λ2))dt + (λ∗1 − λ1)dz1 + (λ∗2 − λ2)dz2

The drift of the exchange rate is completely determined by the interest ratedifferential and the market prices of interest rate risk in the two countries.

408 Construction of a riskless portfolio.409 See Flesaker/Hughston (2000), p. 220.


The price dynamics for the USD money market account and the USDzero-coupon bond in EUR from (5.44) and (5.45) can now be rewritten witha from (5.46) and b1 and b2 from (5.47) and (5.48) as follows

dM∗(t)M∗(t)

=(r1(t) + (λ1 − λ∗1)λ1 + (λ2 − λ∗2)λ2)dt

− (λ1 − λ2)dz1 − (λ2 − λ∗2)dz2 (5.49)

and

dP ∗(t, T )P ∗(t, T )

=(r1(t) + (λ1 − λ∗1)λ1 + (λ2 − λ∗2 + σ2,2(t, T ))λ2)dt

− (λ1 − λ∗1)dz1 − (λ2 − λ∗2 + σ2,2(t, T ))dz2 (5.50)

In this simple example, the instantaneous return of the foreign currency assetsin local currency is completely determined by the local short rate r1 since weassumed constant market prices of interest rate risk.

5.4.3 Derivation of the Optimum Portfolio Weights

The investor can invest his initial wealth in a local zero-coupon bond of ma-turity T1, a foreign zero-coupon bond of maturity T2 and a riskless moneymarket account that yields the local riskfree rate.

We now solve the international bond portfolio selection problem in ac-cordance with the method outlined at the beginning of the chapter. We useEquation (5.2) with410

σt =(

σ1,1(t, T1) 0λ1 − λ∗1 σ2,2(t, T2) + λ2 − λ∗2

)(5.51)

and obtain the following expression for the wealth dynamics

dW

W= µW (r1, t, T1, T2)dt− σW,1(t)dz1 − σW,2(t)dz2 (5.52)

where

µW (r1, t, T1, T2) =r1 + w1σ1,1(t, T1)λ1 + w2((λ1 − λ∗1)λ1

+ (λ2 − λ∗2 + σ2,2(t, T2))λ2)σW,1(t) =w1σ1,1(t, T1) + w2(λ1 − λ∗1)σW,2(t) =w2(λ2 − λ∗2 + σ2,2(t, T2))

Because of arbitrage-relationships and the characteristics of the Vasicek termstructure model (here: assumption of constant market price of interest rate

410 The German zero-coupon bond has maturity date T1 and the US zero-couponbond T2.


risk), the wealth dynamics depend on only two factors, namely W and r1. Theexchange rate and the short rate in the USA r2 don’t influence the wealthdynamics of the investor.

We solve the problem analogous to the Vasicek case in section 5.3.1. Theuncertainty is driven by one state variable r1 only – hence x(t) = r1(t) –and two uncorrelated Brownian motions z1(t) and z2(t). We hence need twopositions in risky assets and a position in the riskless asset, i.e. k = 1, d =n = 2. Since k = 1, our optimum portfolio will consist of a mean-varianceefficient portfolio and a single hedge portfolio.

We can specify the general portfolio weight equation for a CRRA investorin Equation (5.22) as follows

σ =(

σ1,1(t, T1) 0λ1 − λ∗1 λ2 − λ∗2 + σ2,2(t, T2)

)

λ =(

λ1

λ2

)

β =(σr,1 0

)

We assume the following functional form for the separation function f(r1, t).This is the same assumption as in the section on the Vasicek model.411

f(r1, t) = g(t) exp(v(t)r1)

thenfr1

f= v(t)

The zero-coupon bond volatilities are412

σ1,1(t, T ) = σr1B1(t, T ) = σr1

(1− e−κ1(T−t)

κ1

)(5.53)

σ2,2(t, T ) = σr2B2(t, T ) = σr2

(1− e−κ2(T−t)

κ2

)(5.54)

Inserting the assumption for f in Equation (5.21) leads to

er1v(t)r1g(t) (γ − κ1v(t) + v′(t)) +er1v(t)

2(γ − 1)× (

2(γ − 1)g′(t)− g(t)(γλ2

1 − 2γσr1v(t)λ1 + γλ22

+v(t)(σ2

r1v(t)− 2(γ − 1)θ1κ1

)))= 0 (5.55)

411 See Equation 5.24.412 Each zero-coupon bond price is given by the Vasicek bond price formula and so

the volatilities are the same as in Chapter 5.3.1.


The expression γ − κ1v(t) + v′(t) must hence be zero for all values of r1. Thesolution to v(t) is the same as in the Vasicek case, i.e.

v(t) = γ1− exp(−κ1(T − t))

κ1= γB1(t, T ) (5.56)

With v(t) from (5.56) Equation (5.55) becomes an ODE for the unknownfunction g(t). The solution for g(t) is once again not needed for the derivationof the optimum portfolio weight, we give it therefore only in the Appendix.413

v(t) and g(t) solve Equation (5.55). With the solution for v(t) we obtain theoptimum portfolio weights

w∗ =1

1− γ

(Σ−1σλ

)− γ

1− γΣ−1ΣB1(t, T ) (5.57)

The hedge term can be simplified further and we eventually obtain

w∗ =1

1− γ

(Σ−1σλ

)− γ

1− γB1(t, T )

( 1B1(t,T1)

0

)(5.58)

5.4.4 Interpretation of the Optimum Portfolio Weights

We now have a closer look at the optimum portfolio weights from an economicpoint of view. A general discussion of the influence of the parameters on themean-variance portfolio is difficult. The mean-variance efficient portfolio canbe written as ( λ2λ∗1+λ1(σ2(t,T2)−λ∗2)

σ1(t,T1)(λ2−λ∗2+σ2(t,T2))λ2

λ2−λ∗2+σ2(t,T2)

)(5.59)

Utilizing the relationships from (5.47) and (5.48) we can write this as(

b1λ2+(σ2(t,T2)−b2)λ1σ1(t,T1)(σ2(t,T2)−b2)

λ2σ2(t,T2)−b2

)(5.60)

If we assume positive market prices of interest rate risk and positive volatilitiesof the exchange rate, then both portfolio weights are positive if σ2(t, T2) > b2.

The hedge portfolio in Equation (5.58) consists of a position in the localzero-coupon bond only. This seems counterintuitive at first, but is perfectlyrational. We have stated before414, that the hedge portfolio is formed in sucha way as to provide the best possible hedge against the state variables (herer1), in this case this is a position only in the local zero-coupon bond since theinstantaneous correlation coefficient between dr1 and dP1(t,T )

P1(t,T ) is −1. An addi-tional position in the foreign zero-coupon bond would reduce the (absolute)correlation and therefore diminish the effect of the hedge portfolio.

413 See Appendix C.3 , Equation (C.3).414 See Section 5.2.3 on page 93.


It is interesting to note that we can recover the single currency Vasicek(1977) problem415 as a special case by setting λ2 = 0. Then the weight ofthe second bond in the mean-variance efficient portfolio becomes zero, i.e. aposition in the second zero-coupon bond is only taken when λ2 is not equalto zero.

5.4.5 Numerical Example

We now demonstrate the applicability of the model by means of a numericalexample. We assume the following values for the parameters of the term struc-ture models, the zero-coupon bonds, the foreign exchange rate volatilities andthe investor’s risk aversion and investment horizon:

• Investor– Risk aversion parameter: γ = 0.5– Investment horizon: T = 2

• Term structure– Germany: κ1 = 0.7, σr1 = 0.02, λ1 = λ2 = 0.01– USA: κ2 = 0.8, σr2 = 0.03

• Zero-coupon bond maturities: T1 = T2 = 10• Exchange rate: b1 = b2 = 0.01

We insert the parameters into Equation (5.58) and obtain

w∗ =(

0.08781820.403261

)(5.61)

The resulting portfolio has long positions in both zero-coupon bonds and inthe riskless money market account. This result can be disaggregated as follows

Table 5.5. International bond portfolio selection: Disaggregation of optimum port-folio weights.

Weight Mean-variance Hedge portfolio Overall

Local zero 0.841909 -0.754091 0.0878182

Foreign zero 0.403261 0 0.403261

Local MMA — — 0.508921

These results seem quite plausible but as has been said in the last twosections where we covered the Vasicek and the HW2 models, the optimumportfolio weights are heavily dependent on the parameters, especially the un-observable market prices of interest rate risk.

415 The solution was given in Chapter 5.3.1.

5.5 Summary and Conclusion 113

5.5 Summary and Conclusion

The dynamic portfolio selection framework is quite flexible and analytical so-lutions can be derived for simple interest rate models. We derived the optimumportfolio strategy for CRRA investors in the Vasicek and HW2 case. The re-sults for the Vasicek model were already published in Korn/Kraft (2002) butthe specific results for the HW2 model were new.416 We have shown that theframework can be extended to foreign exchange risks. In the numerical ex-amples the resulting portfolio weights unfortunately displayed huge long andshort positions in the assets. It is highly unlikely that these extreme positions(in terms of position size) would ever be implemented in practice.

The results mirror the outcomes of the unrestricted static portfolio selec-tion in Chapter 4. There, we could improve the plausibility of the portfoliosby introducing short-sale constraints. Constrained optimization problems incontinuous-time have been studied by Cvitanic/Karatzas (1992), Xu/Shreve(1992a) and Xu/Shreve (1992b). For more complicated (multi-factor) modelsor constrained optimization problems, numerical methods must be used.417

416 Munk/Sørensen (2004) solve the dynamic optimization problem for HJM termstructure movements. The HW2 model is a special case of the HJM framework,so the optimum portfolio weights could have been recovered from them.

417 For a textbook treatment see Kushner/Dupuis (2000).

6

Summary and Conclusion

The aim of this thesis was to analyze the necessary adjustments to static anddynamic models of equity portfolio optimization in order to use these modelsfor the selection of bond portfolios. Furthermore, we wanted to compare theoptimum portfolios with bond portfolios recommended by the practice. Thecore of this thesis is based mainly upon Wilhelm (1992) and Korn/Kraft(2002).

We derived the general HJM framework for term structure modeling inChapter 3. The derivation employed the stochastic discount factor approachand not the normally utilized martingale or PDE approaches. Then we ob-tained the Vasicek (1977) and the HW2 models as special cases of the generalHJM framework.

Chapter 4 dealt with static bond portfolio optimization methods. We pre-sented the bond portfolio selection model by Wilhelm (1992) and showed indetail how the necessary parameters could be derived. Wilhelm (1992) showedhow to use this model in conjunction with the CIR term structure model. Weextended his approach to the Vasicek and the HW2 models. We found thatwhen the number of assets is far greater than the number of risk factors, themean-variance efficient portfolios contained huge long and short positions.This finding hence confirms the conclusion reached by Korn/Koziol (2006).It is very unlikely that these portfolios could be implemented in practice.Korn/Koziol (2006) suggest limiting the number of bonds to include in theportfolio. We used short-sale constrained optimization to find optimum bondportfolios in this thesis. A drawback of this approach is that no analytic so-lutions can be found for the bond portfolio selection problem, so we had toconfine our analysis to numerical examples. The resulting portfolios containedoftentimes only positions in relatively few zero-coupon bonds. The comparisonof mean-variance efficient portfolios with portfolio propagated by the practiceproduced interesting results. Portfolios resulting from the duration and yieldcurve strategies could not in all cases be reproduced with the mean-varianceframework. It seemed that particular portfolios of bonds are preferred regard-less of the expectations about future term structure movements. Obviously,

116 6 Summary and Conclusion

the main driver for these differences, is taking into consideration the risksof these strategies in the mean-variance framework. On the other hand, theduration-immunized portfolios performed quite well in the Vasicek and theHW2 term structure models. Hence, the simple portfolio rule for implement-ing these portfolios can be seen as a reasonable approximation to reality.

In Chapter 5, we presented dynamic models of bond portfolio selection.These are based on the seminal paper by Merton (1969). His approach can begeneralized to interest rate sensitive assets as well. We presented the optimumportfolio strategy for CRRA investors in a Vasicek model. This result was ob-tained previously by Korn/Kraft (2002). Then we showed how to obtain anexplicit solution for a bond portfolio selection problem with the HW2 modelusing the traditional stochastic control approach. This result presents an ad-dition to the bond portfolio literature.418 We furthermore analyzed a simpleextension of this framework, an international bond portfolio selection prob-lem. We were able to derive an analytic expression for the optimum portfoliostrategy in a two country case. Unfortunately, the unconstrained portfoliosgenerally contained large long and short positions. This finding is similar tothe results we obtained from static portfolio optimization. Constraining shortsales seems to be the obvious measure, but then the resulting optimizationproblem is far more complicated. We refrained from solving constrained dy-namic optimization problems in this thesis.

An application of the models in practice is in our opinion difficult. At themoment, I think, virtually no mutual fund company employs sophisticateddynamic optimization models for (bond) portfolio selection. Hence, an imple-mentation of the static mean-variance model for bond portfolios seems moreprobable. There is a good chance that quantitatively oriented bond portfoliomanagers give this approach a try. But before even those portfolio managersconsider an application of the model, a backtesting has to be performed.419

For an implementation, we recommend a two-factor term structure model,e.g. the HW2 model. Principal component analysis of the term structureshowed that two to three factors account for the vast majority of the vari-ation in the term structure.420 A two-factor model produces also realisticcorrelations between bonds of different maturities.421 A problem that mightarise is the limited number of bonds the mean-variance approach typicallyadvises to buy. Hence, the typical government bond portfolio would consistof only a handful of assets. This concentration of the portfolio value in just acouple of assets might make the client nervous and furthermore (if the fundhas significant money to invest) might affect the prices in the bond market.This is usually refered to as the large-trader problem.422 Another obstacle is

418 Munk/Sørensen (2004) derived optimum bond portfolios for HJM models usingthe martingale approach.

419 This has already been examined for the German market by Korn/Koziol (2006).420 Litterman/Scheinkman (1991).421 In a one-factor model, all bond prices are perfectly correlated.

6 Summary and Conclusion 117

the prevailing relative portfolio management approach, i.e. the managementof a portfolio relative to a benchmark. Mean-variance efficient portfolios maydeviate severely from the composition (and possibly from the risk character-istics) of the benchmark, hence these portfolios will probably have a largetracking error.423

Despite these difficulties, we think this approach is promising. It is basedon an accepted theory and is widely employed in equity portfolio selection.In our opinion it is better suited to cope with the complex fixed income risksthan the ad hoc approaches currently employed in practice.

422 For consumption and investment problems with a large trader, see Cuoco/Cvitanic (1998) and Bank/Baum (2004).

423 The investment guidelines might limit the tracking error of the portfolio.

A

Heath/Jarrow/Morton (1992)

A.1 Dynamics of Zero-Coupon Bonds

The zero-coupon bond price can be written as a function of the instantaneousforward rate curve as follows1

P (t, T ) = exp

(−

∫ T

t

f(t, u) du

)(A.1)

Let’s define Y (t) =∫ T

tf(t, u) du. The problem is now to determine the dy-

namics of Y (t) given the dynamics of the forward rate curve in (3.2). It hasbeen shown by Heath/Jarrow/Morton (1992) by using a generalized stochasticFubini theorem, that the dynamics of Y can be written as2

dY (t) =

[(∫ T

t

m(t, u) du

)− f(t, t)

]dt+

d∑

i=1

(∫ T

t

si(t, u) du

)dzi(t) (A.2)

Since P (t, T ) = g(Y (t)) with g(Y ) = exp(−Y ), g′(Y ) = − exp(−Y ) andg′′(Y ) = exp(−Y ) it follows by application of Ito’s lemma that the dynamicsof P (t, T ) are

dP (t, T ) = g′(Y )dY +12g′′(Y )(dY )2

= − exp(−Y )dY +12

exp(−Y )(dY )2

= −P (t, T )dY +12P (t, T )(dY )2

Inserting dY and recognizing that

1 See Equation (2.10).2 See Heath/Jarrow/Morton (1992), p. 99.

120 A Heath/Jarrow/Morton (1992)

(dY )2 =d∑

i=1

(∫ T

t

si(t, u) du

)2

dt

we obtain the following formula for the evolution of P (t, T )

dP (t, T )P (t, T )

=

f(t, t)−

(∫ T

t

m(t, u) du

)+

12

d∑

i=1

(∫ T

t

si(t, u) du

)2 dt

−d∑

i=1

(∫ T

t

si(t, u) du

)dzi(t)dt (A.3)

A.2 Arbitrage-Free Pricing

Assume that x = ln(ζ), then it follows from Equation (3.11) and an applicationof Ito’s lemma that

dx = xtdt + xζdζ +12xζζ(dζ)2

with

xt =∂x

∂t= 0

xζ =∂x

∂ζ=

1ζ

xζζ =∂2x

∂ζ2= − 1

ζ2

(dζ)2 = ζ(t)2d∑

i=1

λi(t)2dt

Inserting these expressions in dx yields

dx = −f(t, t)dt− 12

d∑

i=1

λi(t)2dt +d∑

i=1

λi(t)dzi(t) (A.4)

Since

x(T ) = x(t) +∫ T

t

dx(u)du

it follows that

x(T ) = x(t) +∫ T

t

−f(u, u)du−∫ T

t

12

d∑

i=1

λi(u)2du +∫ T

t

d∑

i=1

λi(u)dzi(u)

= x(t) +∫ T

t

−f(u, u)du−d∑

i=1

∫ T

t

12λi(u)2du +

d∑

i=1

∫ T

t

λi(u)dzi(u)

A.3 HJM Drift Condition 121

We insert x and apply the exponential function to both sides of the equation.Hence we obtain

ζ(T ) = ζ(t) exp

(∫ T

t

−f(u, u)du +d∑

i=1

∫ T

t

λi(u)dzi(u)−d∑

i=1

∫ T

t

12λi(u)2du

)

A.3 HJM Drift Condition

The dynamics of Y (t, T ) = ζ(t)P (t, T ) can be obtained by a straightforwardapplication of Ito’s lemma with dP (t, T ) from (3.5) and dζ(t) from (3.11).According to Ito’s lemma the dynamics of Y are

dY = Ytdt + Yζdζ + YP dP + YζP (dζ)(dP ) +12Yζζ(dζ)2 +

12YPP (dP )2

with

Yt = 0Yζ = P

YP = ζ

YζP = 1Yζζ = 0

YPP = 0

(dζ)(dP ) = −Pζ

d∑

i=1

σi(t, T )λi(t)dt

We insert these expressions in dY and obtain

dY (t, T ) =P (t, T )

[−f(t, t)ζ(t)dt + ζ(t)

d∑

i=1

λi(t)dzi(t)

]

+ ζ(t)

[P (t, T )µ(t, T )− P (t, T )

d∑

i=1

σi(t, T )dzi(t)

]

− P (t, T )ζ(t)d∑

i=1

σi(t, T )λi(t)dt

Replacing ζ(t)P (t, T ) with Y and simplification yields

dY (t, T )Y (t, T )

=

(−f(t, t) + µ(t, T )−

d∑

i=1

σit, Tλi(t)

)dt

+d∑

i=1

(λi(t)− σi(t, T ))dzi(t) (A.5)

122 A Heath/Jarrow/Morton (1992)

A.4 Special Case: Hull/White (1994)

The function g(0, T ) is defined as follows

g(0, T ) =

− e−2Tκr(−1 + 2eTκr

)κεσ

2r

κr (κr − κε)2 −

e−2Tκrκε

((−1 + eTκr)2

κε − 2e2Tκrκr

)σ2

r

2κ2r (κr − κε)

2

− e−2Tκr(−1 + eTκr

)2σ2

r

2 (κr − κε)2 +

e−Tκr(−1 + eTκr

)λ1σr

κr

+θ − e−Tκrθ

κr+

e−2T (κr+κε)(−1 + eTκr

)

κ2r (κr − κε)κε

×(%

(e2Tκε

(−1 + eTκr)κε − eT (κr+κε)

(−1 + eTκε)κr

)σεσr

)

− e−2T (κr+κε)(eTκr

(−1 + eTκε)κr − eTκε

(−1 + eTκr)κε

)2σ2

ε

2κ2r (κr − κε)

2κ2

ε

+e−T (κr+κε)%

(eTκr

(−1 + eTκε)κr − eTκε

(−1 + eTκr)κε

)λ1σε

κr (κr − κε)κε

+e−2T (κr+κε)

√1− %2

κr (κr − κε)κε

×(eT (2κr+κε)

(−1 + eTκε)κr − eT (κr+2κε)

(−1 + eTκr)κε

)λ2σε (A.6)

B

Dynamic Bond Portfolio Optimization

The calculation of the expected values necessary for the derivation of the HJBequation is shown in detail in this appendix.

E(∆x∆x′) = (α∆t + β∆z)(α∆t + β∆z)′

= (α∆t + β∆z)(α′∆t + ∆z′β′)= β∆z∆z′β′

= βI∆tβ′

= ββ′∆t

E(∆x∆W ) = (α∆t + β∆z)((r + w′σλ)W∆t)− w′σW∆z)= (β∆z)(w′σ∆z)(−W )= (β∆z)(∆z′σ′w)(−W )= βI∆tσ′w(−W )= −βσ′wW

E(∆W 2) = (w′σ∆z(−W ))(w′σ∆z(−W ))= (w′σ∆z(−W ))(∆z′σ′w(−W ))

= w′σ∆z∆z′σ′wW 2

= w′σI∆tσ′wW 2

= w′σσ′wW 2∆t

C

Dynamic Bond Portfolio Optimization

C.1 Vasicek (1977)

The function g(t) is implicitly defined by the following equation

ln(g(t)) =e−2Tκ

(e2tκ + 3e2Tκ − 4e(t+T )κ

)σ2γ2

4(γ − 1)κ3

− e−2Tκσ(e2Tκ(2λ + (T − t)σ)− 2e(t+T )κλ

)γ2

2(γ − 1)κ2+ (T − t)θγ

+(t− T )λ(κλ− 2γσ)γ

2(γ − 1)κ+

(−1 + e(t−T )κ)θγ

κ(C.1)

C.2 Hull/White (1994)


126 C Dynamic Bond Portfolio Optimization

ln(g(t)) =

Tθκ2εγ

2

(γ − 1)κr (κr − κε)2 −

θκ2εγ

2

(γ − 1)κ2r (κr − κε)

2

+(2Tκr − 3) (2κr − κε)κεσ

2rγ2

4(γ − 1)κ3r (κr − κε)

2 +(3− 2Tκr)σ2

rγ2

4(γ − 1)κr (κr − κε)2

+

((3− 2Tκε)κ2

r + κε (5− 2Tκε) κr + 3κ2ε

)σ2

ε γ2

4(γ − 1)κ3rκ

3ε (κr + κε)

+2θκεγ

2

(γ − 1)κr (κr − κε)2

+(κr (Tκε − 1)− κε)

(%λ1 +

√1− %2λ2

)σεγ

2

(γ − 1)κ2rκ

2ε

+%

((2− 2Tκε)κ2

r − 2κε (Tκε − 2)κr + 3κ2ε

)σrσεγ

2

2(γ − 1)κ3rκ

2ε (κr + κε)

− Tθκ2εγ

(γ − 1)κr (κr − κε)2 +

θκ2εγ

(γ − 1)κ2r (κr − κε)

2 −2θ (T (γ − 1)κr + 1) κεγ

(γ − 1)κr (κr − κε)2

+

(−T(λ2

1 + λ22

)κ2

r + 2Tγλ1σrκr − 2γλ1σr

)γ

2(γ − 1)κ2r

− θγ

(κr − κε)2

+Tθκrγ

(κr − κε)2 (C.2)

C.3 International Bond Portfolio Selection


ln(g(t)) =

e−2Tκ1(e2tκ1 + 3e2Tκ1 − 4e(t+T )κ1

)σ2

r1γ2

4(γ − 1)κ31

+(T − t)λ1σr1γ

2

(γ − 1)κ1− λ1σr1γ

2

(γ − 1)κ21

+e−2Tκ1σr1

(2e(t+T )κ1λ1 − e2Tκ1(T − t)σr1

)γ2

2(γ − 1)κ21

+ (T − t)θ1γ

+e−Tκ1

(2

(etκ1 − eTκ1

)(γ − 1)θ1 − eTκ1(T − t)κ1

(λ2

1 + λ22

))γ

2(γ − 1)κ1(C.3)

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List of Tables

3.1 Vasicek (1977) model: Special case of HJM. . . . . . . . . . . . . . . . . . . 223.2 Hull/White (1994) model: Special case of HJM. . . . . . . . . . . . . . . 233.3 Vasicek model: Probability of negative future short rates. . . . . . . 27

4.1 Bond portfolio optimization: Input parameters. . . . . . . . . . . . . . . . 474.2 Vasicek model: Parameter values for numerical example. . . . . . . . 524.3 Vasicek model: Distribution of zero-coupon bond prices. . . . . . . . 544.4 Vasicek model: Expected holding period returns. . . . . . . . . . . . . . 554.5 Vasicek model: Zero-coupon bond returns in different scenarios. 554.6 Vasicek model: Zero-coupon bond weights for short-sale

constrained portfolios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7 Vasicek model: Standard deviations of terminal wealth for

unconstrained and short-sale constrained portfolios . . . . . . . . . . . 574.8 Vasicek model: Zero-coupon bond weights for short-sale

constrained portfolios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.9 Vasicek model: Standard deviations of terminal wealth for

unconstrained and short-sale constrained portfolios . . . . . . . . . . . 604.10 HW2 model: Parameter values for numerical example. . . . . . . . . . 614.11 HW2 model: Expected holding period returns of zero-coupon

bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.12 HW2 model: Zero-coupon bond weights for short-sale

constrained portfolios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.13 HW2 model: Standard deviations of terminal wealth for

unconstrained and short-sale constrained portfolios . . . . . . . . . . . 644.14 HW2 model: Zero-coupon bond weights for short-sale

constrained portfolios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.15 HW2 model: Standard deviations of terminal wealth for

unconstrained and short-sale constrained portfolios . . . . . . . . . . . 654.16 Vasicek model: Riding strategy parameter values. . . . . . . . . . . . . . 68

136 List of Tables

4.17 Vasicek model: Portfolio weights when term structure isexpected to remain unchanged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.18 Vasicek model: Duration strategy parameter values. . . . . . . . . . . . 704.19 Vasicek model: Portfolio weights and durations for duration

strategy comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.20 HW2 model: Duration strategy parameter values. . . . . . . . . . . . . 724.21 HW2 model: Portfolio weights and Macaulay durations for

duration strategy comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.22 Vasicek model: Zero-coupon bond weights when a flattening is

expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.23 HW2 model: Zero-coupon bond weights when a flattening is

expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.24 Vasicek model: Portfolio weights of minimum-variance

portfolios under short-sale constraints for different investmenthorizons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.25 Vasicek model: Comparison of minimum-variance portfoliostandard deviations to duration-immunized portfolios. . . . . . . . . . 79

4.26 HW2 model: Portfolio weights of minimum-variance portfoliosunder short-sale constraints for different investment horizons. . . 80

4.27 HW2 model: Comparison of minimum-variance portfoliostandard deviations to duration-immunized portfolios. . . . . . . . . . 81

4.28 Comparison with real-world portfolio selection methods. . . . . . . . 82

5.1 Existing literature on continuous-time bond portfolio selection. . 865.2 Vasicek model: Zero-coupon bond weight for different market

prices of interest rate risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 HW2 model: Disaggregation of optimum portfolio. . . . . . . . . . . . . 1045.4 HW2 model: Effect of market prices of interest rate risk on

portfolio weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.5 International bond portfolio selection: Disaggregation of

optimum portfolio weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

List of Figures

3.1 Vasicek model: ∂R∂r as a function of maturity. . . . . . . . . . . . . . . . . . 28

3.2 Vasicek model: Volatility structure . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 HW2 model: ∂R

∂r as a function of maturity. . . . . . . . . . . . . . . . . . . . 373.4 HW2 model: ∂R

∂ε as a function of maturity. . . . . . . . . . . . . . . . . . . . 373.5 HW2 model: Volatility structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Vasicek: Term structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Vasicek: Volatility structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 HW2: Term structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4 HW2: Volatility structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Vasicek model: Expected term structure . . . . . . . . . . . . . . . . . . . . . 714.6 Archetypical bullet portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.7 Archetypical barbell portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.8 Archetypical ladder portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1 Vasicek Bond Portfolio Selection: Zero-Coupon Bond weightw as a function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bond portfolio optimization

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